14.3E: Exercises
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Practice Makes Perfect
Exercise \(\PageIndex{17}\) Determine if a Sequence is Arithmetic
In the following exercises, determine if each sequence is arithmetic, and if so, indicate the common difference.
- \(4,12,20,28,36,44, \dots\)
- \(-7,-2,3,8,13,18, \dots\)
- \(-15,-16,3,12,21,30, \dots\)
- \(11,5,-1,-7-13,-19, \dots\)
- \(8,5,2,-1,-4,-7, \dots\)
- \(15,5,-5,-15,-25,-35, \dots\)
- Answer
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1. The sequence is arithmetic with common difference \(d=8\).
3. The sequence is not arithmetic.
5. The sequence is arithmetic with common difference \(d=−3\).
Exercise \(\PageIndex{18}\) Determine if a Sequence is Arithmetic
In the following exercises, write the first five terms of each sequence with the given first term and common difference.
- \(a_{1}=11\) and \(d=7\)
- \(a_{1}=18\) and \(d=9\)
- \(a_{1}=-7\) and \(d=4\)
- \(a_{1}=-8\) and \(d=5\)
- \(a_{1}=14\) and \(d=-9\)
- \(a_{1}=-3\) and \(d=-3\)
- Answer
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1. \(11,18,25,32,39\)
3. \(-7,-3,1,5,9\)
5. \(14,5,-4,-13,-22\)
Exercise \(\PageIndex{19}\) Find the General Term (\(n\)the Term) of an Arithmetic Sequence
In the following exercises, find the term described using the information provided.
- Find the twenty-first term of a sequence where the first term is three and the common difference is eight.
- Find the twenty-third term of a sequence where the first term is six and the common difference is four.
- Find the thirtieth term of a sequence where the first term is \(−14\) and the common difference is five.
- Find the fortieth term of a sequence where the first term is \(−19\) and the common difference is seven.
- Find the sixteenth term of a sequence where the first term is \(11\) and the common difference is \(−6\).
- Find the fourteenth term of a sequence where the first term is eight and the common difference is \(−3\).
- Find the twentieth term of a sequence where the fifth term is \(−4\) and the common difference is \(−2\). Give the formula for the general term.
- Find the thirteenth term of a sequence where the sixth term is \(−1\) and the common difference is \(−4\). Give the formula for the general term.
- Find the eleventh term of a sequence where the third term is \(19\) and the common difference is five. Give the formula for the general term.
- Find the fifteenth term of a sequence where the tenth term is \(17\) and the common difference is seven. Give the formula for the general term.
- Find the eighth term of a sequence where the seventh term is \(−8\) and the common difference is \(−5\). Give the formula for the general term.
- Find the fifteenth term of a sequence where the tenth term is \(−11\) and the common difference is \(−3\). Give the formula for the general term.
- Answer
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1. \(163\)
3. \(131\)
5. \(-79\)
7. \(a_{20}=-34 .\) The general term is \(a_{n}=-2 n+6\).
9. \(a_{11}=59 .\) The general term is \(a_{n}=5 n+4\).
11. \(a_{8}=-13 .\) The general term is \(a_{n}=-5 n+27\).
Exercise \(\PageIndex{20}\) Find the General Term (\(n\)the Term) of an Arithmetic Sequence
In the following exercises, find the first term and common difference of the sequence with the given terms. Give the formula for the general term.
- The second term is \(14\) and the thirteenth term is \(47\).
- The third term is \(18\) and the fourteenth term is \(73\).
- The second term is \(13\) and the tenth term is \(−51\).
- The third term is four and the tenth term is \(−38\).
- The fourth term is \(−6\) and the fifteenth term is \(27\).
- The third term is \(−13\) and the seventeenth term is \(15\).
- Answer
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1. \(a_{1}=11, d=3 .\) The general term is \(a_{n}=3 n+8\).
3. \(a_{1}=21, d=-8 .\) The general term is \(a_{n}=-8 n+29\)
5. \(a_{1}=-15, d=3 .\) The general term is \(a_{n}=3 n-18\).
Exercise \(\PageIndex{21}\) Find the Sum of the First \(n\) Terms of an Arithmetic Sequence
In the following exercises, find the sum of the first \(30\) terms of each arithmetic sequence.
- \(11,14,17,20,23, \dots\)
- \(12,18,24,30,36, \dots\)
- \(8,5,2,-1,-4, \dots\)
- \(16,10,4,-2,-8, \dots\)
- \(-17,-15,-13,-11,-9, \dots\)
- \(-15,-12,-9,-6,-3, \dots\)
- Answer
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1. \(1,635\)
3. \(-1,065\)
5. \(360\)
Exercise \(\PageIndex{22}\) Find the Sum of the First \(n\) Terms of an Arithmetic Sequence
In the following exercises, find the sum of the first \(50\) terms of the arithmetic sequence whose general term is given.
- \(a_{n}=5 n-1\)
- \(a_{n}=2 n+7\)
- \(a_{n}=-3 n+5\)
- \(a_{n}=-4 n+3\)
- Answer
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1. \(6,325\)
3. \(-3,575\)
Exercise \(\PageIndex{23}\) Find the Sum of the First \(n\) Terms of an Arithmetic Sequence
In the following exercises, find each sum.
- \(\sum_{i=1}^{40}(8 i-7)\)
- \(\sum_{i=1}^{45}(7 i-5)\)
- \(\sum_{i=1}^{50}(3 i+6)\)
- \(\sum_{i=1}^{25}(4 i+3)\)
- \(\sum_{i=1}^{35}(-6 i-2)\)
- \(\sum_{i=1}^{30}(-5 i+1)\)
- Answer
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1. \(6,280\)
3. \(4,125\)
5. \(-3,580\)
Exercise \(\PageIndex{24}\) Writing Exercises
- In your own words, explain how to determine whether a sequence is arithmetic.
- In your own words, explain how the first two terms are used to find the tenth term. Show an example to illustrate your explanation.
- In your own words, explain how to find the general term of an arithmetic sequence.
- In your own words, explain how to find the sum of the first \(n\) terms of an arithmetic sequence without adding all the terms.
- Answer
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1. Answer may vary
3. Answer may vary
Self Check
a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
b. After reviewing this checklist, what will you do to become confident for all objectives?