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If it is $0$, then $n$ is a multiple of $3$. Write the first five terms of the arithmetic sequence. Theory of Equations 3. A. Similarly to above, since $n^5-n$ is divisible by $n-1$, $n$, and $n+1$, it must have a factor which is a multiple of $3$, and therefore must itself be divisible by $3$. A geometric series22 is the sum of the terms of a geometric sequence. 14) a1 = 1 and an + 1 = an for n 1 15) a1 = 2 and an + 1 = 2an for n 1 Answer 16) a1 = 1 and an + 1 = (n + 1)an for n 1 17) a1 = 2 and an + 1 = (n + 1)an / 2 for n 1 Answer If #lim_{n->infty}|a_{n+1}|/|a_{n}| < 1#, the Ratio Test will imply that #sum_{n=1}^{\infty}a_{n}=sum_{n=1}^{infty}n/(5^(n))# converges. A certain ball bounces back to one-half of the height it fell from. This might lead to some confusion as to why exactly you missed a particular question. Show that, for every real number y, there is a sequence of rational numbers which converges to y. Use $r = 2$ and the fact that $a_{1} = 4$ to calculate the sum of the first $10$ terms, $\begin{aligned} S_{n} &=\frac{a_{1}\left(1-r^{n}\right)}{1-r} \\ S_{10} &=\frac{\color{Cerulean}{4}\color{black}{\left[1-(\color{Cerulean}{-2}\color{black}{)}^{10}\right]}}{1-(\color{Cerulean}{-2}\color{black}{)}} ] \\ &=\frac{4(1-1,024)}{1+2} \\ &=\frac{4(-1,023)}{3} \\ &=-1,364 \end{aligned}$. The t Write a formula for the general term or nth term for the sequence. https://mathworld.wolfram.com/FibonacciNumber.html, https://www.calculatorsoup.com/calculators/discretemathematics/fibonacci-calculator.php. where $a_{1} = 18$ and $r = \frac{2}{3}$. 2, 0, -18, -64, -5, Find the next two terms of the given sequence. Give an example of a sequence that is arithmetic and a sequence that is not arithmetic. Sequences Quiz Review If it converges, find the limit. List the first five terms of the sequence. Find the sum of the infinite geometric series: a) \sum\limits_{n=0}^\infty \left(\frac{1}{2} \right) ^n . n however, it could be easier to find Fn and solve for Assume that the first term in the sequence is a_1: \{\frac{3}{4}, \frac{4}{9}, \frac{5}{16}, \frac{6}{25}, \}. Was immer er auch probiert, um seinen unverwechselbaren Platz im Rudel zu finden - immer ist ein anderer geschickter, klger What is the 18th term of the following arithmetic sequence? a_n = (1 + 7 / n)^n. Solve for $a_{1}$ in the first equation, $-2=a_{1} r \quad \Rightarrow \quad \frac{-2}{r}=a_{1}$ The first 15 numbers in the sequence, from F0 to F14, are, 0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377. Access the answers to hundreds of Sequences questions that are explained in a way that's easy for you to understand. Web27 Questions Show answers. Complete the next two equations of this sequence: 1 = 1 \\1 - 4 = 3 \\1 - 4 + 9 = 6 \\1 - 4 + 9 - 16 = - 10. To find the common difference between two terms, is taking the difference and dividing by the number of terms a viable workaround? This expression is also divisible by $5$, although this is slightly tricker to show than in the previous two parts. (Assume n begins with 1.). a1 = 1 a2 = 1 an = an 1 + an 2 for n 3. If it converges, find the limit. If (an) is an increasing sequence and (bn) is a sequence of positive real numbers, then (an.bn) is an increasing sequence. Direct link to Timber Lin's post warning: long answer On day three, the scientist observes 17 cells in the sample and Write the first six terms of the arithmetic sequence. Answer 1, contains which literally means doing buying thing, in other words do shopping.if(typeof ez_ad_units!='undefined'){ez_ad_units.push([[250,250],'jlptbootcamp_com-box-4','ezslot_7',105,'0','0'])};__ez_fad_position('div-gpt-ad-jlptbootcamp_com-box-4-0'); Answer 2, contains which means going for a walk. Helppppp will make Brainlyist y is directly proportional to x^2. {1/5, -4/11, 9/17, -16/23, }. Let a_n = \frac{(1 + \sqrt{5})^n - (1 - \sqrt{5})^n}{2^n \sqrt{5}} be a sequence with nth term an. sequence a. Button opens signup modal. WebPre-Algebra. If the limit does not exist, then explain why. WebSolution For Here are the first 5 terms of an arithmetic sequence.3,1,5,9,13Find an expression, in terms of n, for the nth term of this sequence. Explain arithmetic progression and geometric progression. Sequences have many applications in various mathematical disciplines due to their properties of convergence. Direct link to Tzarinapup's post The reason we use a(n)= a, Posted 6 years ago. Permutation & Combination 6. If a_n is a sequence and limit (n tends to infinity) a_n = infinity, then the sequence diverges. Give two examples. &=n(n-1)(n+1)(n^2+1). Sketch a graph that represents the sequence: 7, 5.5, 4, 2.5, 1. \{ \frac{1}{4}, \frac{-2}{9}, \frac{3}{16}, \frac{-4}{25}, \}, Find a formula for the general term and of the sequence, assuming that the pattern of the first few terms continues. Can you figure out the next few numbers? A golf ball bounces back off of a cement sidewalk three-quarters of the height it fell from. Notice the use of the particle here. Simplify (5n)^2. Extend the series below through combinations of addition, subtraction, multiplication and division. $1-\left(\frac{1}{10}\right)^{6}=1-0.00001=0.999999$. -n is even, F-n = -Fn. A geometric series22 is the sum of the terms of a geometric sequence. a_n = \dfrac{5 -2n}{3n -7}, A definite relationship exists among the numbers in the series. Extend the series below through combinations of addition, subtraction, multiplication and division. (Assume n begins with 1.) {(-1)^n}_{n = 0}^infinity. a. Determinants 9. (Assume n begins with 1.) }}, Write the first five terms of the sequence. Find the limit of the sequence: a_n = 2n/(3n + 1). Find an equation for the general term of the given geometric sequence and use it to calculate its $10^{th}$ term: $3, 6, 12, 24, 48$. https://www.calculatorsoup.com - Online Calculators. The pattern is continued by adding 3 to the last number each time, like this: This sequence has a difference of 5 between each number. Walking is usually not considered working. a_1 = -y, d = 5y, Find the first 10 terms of the sequence. \{\frac{n! The common You can view the given recurrent sequence in this way: The $(n+1)$-th term is the average of $n$-th term and $5$. Find a formula for the general term a_n of the sequence \displaystyle{ \{a_n\}_{n=1}^\infty = \left\{1, \dfrac{ 5}{2}, \dfrac{ 25}{4}, \dfrac{ 125}{8}, \dots \right\} } as Find the limit of the sequence whose terms are given by a_n = (n^2) (1 - cos (1.8 / n)). Compute the first five terms of the sequence using the format for a dynamical system defined by a difference equation: Delta t_n = 1.5(100 - t_n), t_0 = 200. List the first four terms of the sequence whose nth term is a_n = (-1)^n + 1 / n. Solve the recurrence relation a_n = 2a_n-1 + 8a_n-2 with initial conditions a_0 = 1, a_1 = 4. Given the geometric sequence, find a formula for the general term and use it to determine the $5^{th}$ term in the sequence. Answer 4, means to enter, but this usually means to enter a room and not a vehicle. Now, let's consider the total number of possible recognition sites. a_n = (-2)^{n + 1}. &=5(5m^2+6m+2). So this is one minus 4/1 plus six. Find a rule for this arithmetic sequence. Suppose a_n is an always increasing sequence. Write a formula that gives the number of cells after any $4$-hour period. If arithmetic or geometric, find t(n). a) Find the nth term. n^2+1&=(5k+2)^2+1\\ Free PDF Download Vocabulary From Classical Roots A Grade Well, means the day before yesterday, and is noon. a_n = (1+3/n)^n. In general, given the first term $a_{1}$ and the common ratio $r$ of a geometric sequence we can write the following: $\begin{aligned} a_{2} &=r a_{1} \\ a_{3} &=r a_{2}=r\left(a_{1} r\right)=a_{1} r^{2} \\ a_{4} &=r a_{3}=r\left(a_{1} r^{2}\right)=a_{1} r^{3} \\ a_{5} &=r a_{3}=r\left(a_{1} r^{3}\right)=a_{1} r^{4} \\ & \vdots \end{aligned}$. copyright 2003-2023 Homework.Study.com. If it converges, find the limit. Show directly from the definition that the sequence \left ( \frac{n + 1}{n} \right ) is a Cauchy sequence. Given the terms of a geometric sequence, find a formula for the general term. The first term of a sequence along with a recursion formula for the remaining terms is given below. Then so is $n^5-n$, as it is divisible by $n^2+1$. - True - False. Explicit formulas for arithmetic sequences | Algebra a_n = \frac{2^{n+1}}{2^n +1}. What recursive formula can be used to generate the sequence 5, -1, -7, -13, -19, where f(1) = 5 and n is greater than 1? sequence , 6n + 7. . 21The terms between given terms of a geometric sequence. How do you write the first five terms of the sequence a_n=3n+1? -7, -4, -1, What is the 7th term of the following arithmetic sequence? $-\frac{1}{5}=r$, $\begin{aligned} a_{1} &=\frac{-2}{r} \\ &=\frac{-2}{\left(-\frac{1}{5}\right)} \\ &=10 \end{aligned}$. Though he gained fame as a magician and escape artist. The infinite sum of a geometric sequence can be calculated if the common ratio is a fraction between $1$ and $1$ (that is $|r| < 1$) as follows: $S_{\infty}=\frac{a_{1}}{1-r}$. Determine whether each sequence is arithmetic or not if yes find the next three terms. And is there another term for formulas using the. The reason we use a(n)= a+b( n-1 ), is because it is more logical in algebra. This is the same format you will use to submit your final answers on the JLPT. Write the first five terms of the sequence. If it converges, find the limit. (b) What does it mean to say that \displaystyle \lim_{n \to \infty} a_n = 8? a n = ( 1 ) n 8 n, Find the limit of the following sequence or determine that the limit does not exist please. What is the rule for the sequence 3, 4, 7, 12? a_n = (n^2)/(n^3 + 1). The first term of a geometric sequence may not be given. An explicit formula directly calculates the term in the sequence that you want. In Become a tutor About us Student login Tutor login. Use the pattern to write the nth term of the sequence as a function of n. a_1=81, a_k+1 = 1/3 a_k, Write the first five terms of the sequence. b) \sum\limits_{n=0}^\infty 2 \left(\frac{3}{4} \right)^n . In the sequence -1, -5, -9, -13, (a) Is -745 a term? In order to find the fifth term, for example, we need to plug, We can get any term in the sequence by taking the first term. a_n = \frac{2n}{n + 1}, Use a graphing utility to graph the first 10 terms of the sequence. Thus we have n terms, plus two, when n = 0 and n = -1. $\begin{aligned} S_{n} &=\frac{a_{1}\left(1-r^{n}\right)}{1-r} \\ S_{6} &=\frac{\color{Cerulean}{-10}\color{black}{\left[1-(\color{Cerulean}{-5}\color{black}{)}^{6}\right]}}{1-(\color{Cerulean}{-5}\color{black}{)}} \\ &=\frac{-10(1-15,625)}{1+5} \\ &=\frac{-10(-15,624)}{6} \\ &=26,040 \end{aligned}$, Find the sum of the first 9 terms of the given sequence: $-2,1,-1 / 2, \dots$. If the 2nd term of an arithmetic sequence is -15 and the 7th term is 10, find the 4th term. a_1 = What is the 5^{th} term in the sequence? Consider the sequence 67, 63, 59, 55 Show that the sequence is arithmetic. a_1 = 100, d = -8, Find a formula for a_n for the arithmetic sequence. Then find an expression for the nth partial sum. . If it converges, find the limit. a_(n + 1) = (a_n)^2 - 1; a_1 = 1. As a matter of fact, for all words on the known vocabulary lists for the JLPT, is read as . 1, - \frac{1}{4}, \frac{1}{9}, - \frac{1}{16}, \frac{1}{25}, \cdots (a) a_n = \frac{(-1)^n}{n^2} (b) a_n = \frac{(-1)^{2n + 1}}{n^2} (c) a Find the 66th term in the following arithmetic sequence. $\begin{aligned} S_{15} &=\frac{a_{1}\left(1-r^{15}\right)}{1-r} \\ &=\frac{9 \cdot\left(1-3^{15}\right)}{1-3} \\ &=\frac{9(-14,348,906)}{-2} \\ &=64,570,077 \end{aligned}$, Find the sum of the first 10 terms of the given sequence: $4, 8, 16, 32, 64, $. a_n = (1 over 2)^n (n), Determine if the following sequence is monotone or strictly monotone. a_8 = 26, a_{12} = 42, Write the first five terms of the sequence. What is the dollar amount? Q. i.e. a_n = cot ({n pi} / {2 n + 3}). Let #a_{n}=n/(5^(n))#. Given the geometric sequence defined by the recurrence relation $a_{n} = 6a_{n1}$ where $a_{1} = \frac{1}{2}$ and $n > 1$, find an equation that gives the general term in terms of $a_{1}$ and the common ratio $r$. Mark off segments of lengths 1, 2, 3, . time, like this: This sequence starts at 10 and has a common ratio of 0.5 (a half). Using the nth term - Sequences - Edexcel - BBC Bitesize Find the nth term of the sequence: 2, 6, 12, 20, 30 Clearly the required sequence is double the one we have found the nth term for, therefore the nth term of the required sequence is 2n(n+1)/2 = n(n + 1). a_1 = 6, a_(n + 1) = (a_n)/n. If it converges, find the limit. Find a formula for the general term an of the sequence starting with a1: 4/10, 16/15, 64/20, 256/25,. Find a formula for the general term, a_n. this, Posted 6 years ago. WebThough you will likely need to use a computer to listen to the audio for the listening section.. First, you should download the: blank answer sheet. You might be thinking that is noon and it is, but is slightly more conversational, whereas is more formal or businesslike. Give the formula for the general term. For example, to calculate the sum of the first $15$ terms of the geometric sequence defined by $a_{n}=3^{n+1}$, use the formula with $a_{1} = 9$ and $r = 3$. For the other answers, the actions are taking place at a location () marked by . . {a_n} = {1 \over {3n - 1}}. A geometric sequence18, or geometric progression19, is a sequence of numbers where each successive number is the product of the previous number and some constant $r$. 3, 5, 7, 9, . False, Determine if the following sequence is monotone or strictly monotone. Answer 1, is dark. Exercises for Sequences (a) n + 2 terms, since to get 1 using the formula 6n + 7 we must use n = 1. Rewrite the first five terms of the sequence (a) using the table feature of a graphing utility and (b) algebraically. List the first five terms of the sequence. List the first five terms of the sequence. Introduction B^n = 2b(n -1) when n>1. Then find a_{10}. If la_n| converges, then a_n converges. Create a scatter plot of the terms of the sequence. 9.3: Geometric Sequences and Series - Mathematics LibreTexts Note that the ratio between any two successive terms is $\frac{1}{100}$. If the sequence is not arithmetic or geometric, describe the pattern. If it is $2$, then $n+1$ is a multiple of $3$. Web1st step. This sequence starts at 1 and has a common ratio of 2. a_n=\frac{(n+1)!}{n! How do you use basic comparison test to determine whether the given series converges or diverges See all questions in Direct Comparison Test for Convergence of an Infinite Series. a_n = (2^n)/(2^n + 1). This formula allows us to simply plug in the number of the term we are interested in, and we will get the value of that term. I personally use all of these on a daily basis and highly recommend them. The. an = 3rd root of n / 3rd root of n + 5. $400$ cells; $800$ cells; $1,600$ cells; $3,200$ cells; $6,400$ cells; $12,800$ cells; $p_{n} = 400(2)^{n1}$ cells. All steps. If it is, find the common difference. Such sequences can be expressed in terms of the nth term of the sequence. Basic Math. a_n = \frac{1 + (-1)^n}{n}, Use the table feature of a graphing utility to find the first 10 terms of the sequence. Sequences Write an explicit definition of the sequence and use it to find the 12th term. These kinds of questions will be some of the easiest on the test so take some time and drill the katakana until you have it mastered. Question Find the nth term. WebFind the sum of the first five terms of the sequence with the given general term. The 2 is found by adding the two numbers before it (1+1) The pattern is continued by multiplying by 2 each The answers to today's Quordle Daily Sequence, game #461, are SAVOR SHUCK RURAL CORAL Quordle answers: The past 20 Quordle #460, Saturday 29 Identify the common ratio of a geometric sequence. \end{align*}\], \[\begin{align*} The distances the ball falls forms a geometric series, $27+18+12+\dots \quad\color{Cerulean}{Distance\:the\:ball\:is\:falling}$. Find k given that k-1, 13, and 3k+3 are consecutive terms of an arithmetic sequence. $\frac{2}{125}=a_{1} r^{4}$. In this case, the nth term = 2n. If the ball is initially dropped from $8$ meters, approximate the total distance the ball travels. Therefore, we next develop a formula that can be used to calculate the sum of the first $n$ terms of any geometric sequence. Determine if the sequence {a_n} converges, and if it does, find its limit when a_n = dfrac{6n+(-1)^n}{4n+2}. d_n = 6n + 7 Find d_{204}. What is the sum of the first seven terms of the following arithmetic sequence? Write a Free PDF Download Vocabulary From Classical Roots A Grade If possible, give the sum of the series. Write an expression for the apparent nth term (a_n) of the sequence. Find x. Find a formula for the general term of a geometric sequence. Note that the ratio between any two successive terms is $2$. }, Find a formula for the nth term, an, of the sequence assuming that the indicated pattern continues. What is the 4th term of the sequence? Student Tutor. Complex Numbers 5. Thats because $n-1$, $n$ and $n+1$ are three consecutive integers, so one of them must be a multiple of $3$. True b. false. Assume n begins with 1. a_n = n/(n^2+1), Write the first five terms of the sequence. 1, \frac{1}{4}, \frac{1}{9}, \frac{1}{16}, \frac{1}{25}, Write an expression for the apparent nth term (a_n) of the sequence. The formula for the Fibonacci Sequence to calculate a single Fibonacci Number is: Fn = ( (1 + 5)^n - (1 - 5)^n ) / (2^n 5). Find the common difference in the following arithmetic sequence. For the second section, you need to choose the correct kanji or just for N5 the katakana. -10, -6, -2, What is the sum of the next five terms of the following arithmetic sequence? If this remainder is 1 1, then n1 n 1 is divisible by 5 5, and then so is n5 n n 5 n, as it is divisible by n1 n 1. If this remainder is 2 2, then n n is 2 2 greater than a multiple of 5 5. That is, we can write n =5k+2 n = 5 k + 2 for some integer k k. Then Sequences All rights reserved. Determine whether the sequence converges or diverges. $\begin{aligned} 0.181818 \ldots &=0.18+0.0018+0.000018+\ldots \\ &=\frac{18}{100}+\frac{18}{10,000}+\frac{18}{1,000,000}+\ldots \end{aligned}$. This is read, the limit of $(1 r^{n})$ as $n$ approaches infinity equals $1$. While this gives a preview of what is to come in your continuing study of mathematics, at this point we are concerned with developing a formula for special infinite geometric series. . Write the first five terms of the sequence and find the limit of the sequence (if it exists). What is the nth term of the sequence 2, 5, 10, 17, 26 ? If the limit does not exist, then explain why. $\frac{2}{125}=a_{1} r^{4}$ N5 Sample Questions Vocabulary Section Explained, JLPT Strategies How to Answer Multiple Choice Questions, JLPT BC 139 | Getting Closer to the July Test, JLPT BC 135 | Adding Grammar and Vocabulary Back In, JLPT Boot Camp - The Ultimate Study Guide to passing the Japanese Language Proficiency Test. Solved 6. Show that the sequence {n2n5+2n} diverges to - Chegg Construct a geometric sequence where $r = 1$. Find the fifth term of this sequence. Show all your work/steps. Determine whether the sequence converges or diverges. For example, the following are all explicit formulas for the sequence, The formulas may look different, but the important thing is that we can plug an, Different explicit formulas that describe the same sequence are called, An arithmetic sequence may have different equivalent formulas, but it's important to remember that, Posted 6 years ago. If \lim_{n \to x} a_n = L, then \lim_{n \to x} a_{2n + 1} = L. Determine whether each sequence is arithmetic or not if yes find the next three terms. Explore the $n$th partial sum of such a sequence. Find term 21 of the following sequence. If arithmetic or geometric, find t(n). So $30$ divides every number in the sequence. 1,\, 4,\, 7,\, 10\, \dots. Give two examples. 7 + 14 + 21 + + 98, Determine the sum of the following arithmetic series. Web1 Personnel Training N5 Previous Question Papers Pdf As recognized, adventure as without difficulty as experience more or less lesson, amusement, as .? The equation for calculating the sum of a geometric sequence: Using the same geometric sequence above, find the sum of the geometric sequence through the 3rd term. \frac{2}{3}, \frac{3}{4},\frac{4}{5}, \frac{5}{6}, \frac{6}{7}, Write the first five terms of the sequence. what are the first 4 terms of n+5 - Brainly.in Multiplying both sides by $r$ we can write, $r S_{n}=a_{1} r+a_{1} r^{2}+a_{1} r^{3}+\ldots+a_{1} r^{n}$. In general, $S_{n}=a_{1}+a_{1} r+a_{1} r^{2}+\ldots+a_{1} r^{n-1}$. Fibonacci numbers occur often, as well as unexpectedly within mathematics and are the subject of many studies. \left\{\begin{matrix} a(1)=-11\\ a(n)=a(n-1)\cdot 10 \end{matrix}\right. b. a_1 = 100, a_{25} = 220, n = 25, Write the first five terms of the sequence and find the limit of the sequence (if it exists). WebTitle: 65.pdf Author: Mo Created Date: 5/22/2016 1:00:55 AM a n = cot n 2 n + 3, List the first three terms of each sequence. To determine a formula for the general term we need $a_{1}$ and $r$. Find the limit of s(n) as n to infinity. It is also commonly desirable, and simple, to compute the sum of an arithmetic sequence using the following formula in combination with the previous formula to find an: Using the same number sequence in the previous example, find the sum of the arithmetic sequence through the 5th term: A geometric sequence is a number sequence in which each successive number after the first number is the multiplication of the previous number with a fixed, non-zero number (common ratio). Here are the answers:if(typeof ez_ad_units!='undefined'){ez_ad_units.push([[300,250],'jlptbootcamp_com-medrectangle-4','ezslot_6',115,'0','0'])};__ez_fad_position('div-gpt-ad-jlptbootcamp_com-medrectangle-4-0'); 3) 4 is the correct answer. Assume n begins with 1. a_n = (2n-3)/(5n+4), Write the first five terms of the sequence. Consider the following sequence 15, - 150, 1500, - 15000, 150000, Find the 27th term. (a) What is a sequence? Sequences Questions and Answers | Homework.Study.com In an Arithmetic Progression, the 9th term is 2 times the 4th term and the 12th term is 78. a n = n 3 + n 2 + 1 2 n 3 2 n + 2. If you are looking for a different level of the test I have notes for each level N5, N4, N3, N2, and N1. Does the sequence appear to have a limit? means to serve or to work (for) someone, which has a very similar meaning to (to work). An explicit formula directly calculates the term in the sequence that you want. b. $a_{1}=\frac{3}{4}$ and $a_{4}=-\frac{1}{36}$, $a_{3}=-\frac{4}{3}$ and $a_{6}=\frac{32}{81}$, $a_{4}=-2.4 \times 10^{-3}$ and $a_{9}=-7.68 \times 10^{-7}$, $a_{1}=\frac{1}{3}$ and $a_{6}=-\frac{1}{96}$, $a_{n}=\left(\frac{1}{2}\right)^{n} ; S_{7}$, $a_{n}=\left(\frac{2}{3}\right)^{n-1} ; S_{6}$, $a_{n}=2\left(-\frac{1}{4}\right)^{n} ; S_{5}$, $\sum_{n=1}^{5} 2\left(\frac{1}{2}\right)^{n+2}$, $\sum_{n=1}^{4}-3\left(\frac{2}{3}\right)^{n}$, $a_{n}=\left(\frac{1}{5}\right)^{n} ; S_{\infty}$, $a_{n}=\left(\frac{2}{3}\right)^{n-1} ; S_{\infty}$, $a_{n}=2\left(-\frac{3}{4}\right)^{n-1} ; S_{\infty}$, $a_{n}=3\left(-\frac{1}{6}\right)^{n} ; S_{\infty}$, $a_{n}=-2\left(\frac{1}{2}\right)^{n+1} ; S_{\infty}$, $a_{n}=-\frac{1}{3}\left(-\frac{1}{2}\right)^{n} ; S_{\infty}$, $\sum_{n=1}^{\infty} 2\left(\frac{1}{3}\right)^{n-1}$, $\sum_{n=1}^{\infty}\left(\frac{1}{5}\right)^{n}$, $\sum_{n=1}^{\infty}-\frac{1}{4}(3)^{n-2}$, $\sum_{n=1}^{\infty} \frac{1}{2}\left(-\frac{1}{6}\right)^{n}$, $\sum_{n=1}^{\infty} \frac{1}{3}\left(-\frac{2}{5}\right)^{n}$.

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