# 23.3: Exercises

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## Exercise $$\PageIndex{1}$$

Find the first seven terms of the sequence.

1. $$a_n=3n$$
2. $$a_n=5n+3$$
3. $$a_n=n^2+2$$
4. $$a_n=n$$
5. $$a_n=(-1)^{n+1}$$
6. $$a_n=\dfrac{\sqrt{n+1}}{n}$$
7. $$a_k=10^k$$
8. $$a_i=5+(-1)^i$$
1. $$3, 6, 9, 12, 15, 18, 21$$
2. $$8, 13, 18, 23, 28, 33, 38$$
3. $$3, 6, 11, 18, 27, 38, 51$$
4. $$1, 2, 3, 4, 5, 6, 7$$
5. $$1, −1, 1, −1, 1, −1, 1$$
6. $$\sqrt{2}, \dfrac{\sqrt{3}}{2}, \dfrac{2}{3}, \dfrac{\sqrt{5}}{4}, \dfrac{\sqrt{6}}{5}, \dfrac{\sqrt{7}}{6}, \dfrac{\sqrt{8}}{7}$$
7. $$10, 100, 1000, 10000, 100000, 1000000, 10000000$$
8. $$4, 6, 4, 6, 4, 6, 4$$

## Exercise $$\PageIndex{2}$$

Find the first six terms of the sequence.

1. $$a_1=5$$, $$a_n=a_{n-1}+3$$ for $$n\geq 2$$
2. $$a_1=7$$, $$a_n=10\cdot a_{n-1}$$ for $$n\geq 2$$
3. $$a_1=1$$, $$a_n=2\cdot a_{n-1}+1$$ for $$n\geq 2$$
4. $$a_1=6$$, $$a_2=4$$, $$a_n=a_{n-1}-a_{n-2}$$ for $$n\geq 3$$
1. $$5, 8, 11, 14, 17$$
2. $$7, 70, 700, 7000, 70000$$
3. $$1, 3, 7, 15, 31$$
4. $$6, 4, −2, −6, −4$$

## Exercise $$\PageIndex{3}$$

Find the value of the series.

1. $$\sum_{n=1}^4 a_n$$, where $$a_n=5n$$
2. $$\sum_{k=1}^5 a_k$$, where $$a_k=k$$
3. $$\sum_{i=1}^4 a_i$$, where $$a_n=n^2$$
4. $$\sum_{n=1}^6 (n-4)$$
5. $$\sum_{k=1}^3 (k^2+4k-4)$$
6. $$\sum_{j=1}^4 \dfrac{1}{j+1}$$
1. $$50$$
2. $$15$$
3. $$30$$
4. $$−3$$
5. $$26$$
6. $$\dfrac {77}{60}$$

## Exercise $$\PageIndex{4}$$

Is the sequence below part of an arithmetic sequence? In the case that it is part of an arithmetic sequence, find the formula for the $$n$$th term $$a_n$$ in the form $$a_n=a_1+d\cdot (n-1)$$.

1. $$5, 8, 11, 14, 17, \dots$$
2. $$-10, -7, -4, -1, 2, \dots$$
3. $$-1, 1, -1, 1, -1, 1, \dots$$
4. $$18, 164, 310, 474, \dots$$
5. $$73.4, 51.7, 30, \dots$$
6. $$9, 3, -3, -8, -14, \dots$$
7. $$4, 4, 4, 4, 4, \dots$$
8. $$-2.72, -2.82, -2.92, -3.02, -3.12, \dots$$
9. $$\sqrt{2}, \sqrt{5}, \sqrt{8}, \sqrt{11}, \dots$$
10. $$\dfrac{-3}{5}, \dfrac{-1}{10}, \dfrac{2}{5}, \dots$$
11. $$a_n=4+5\cdot n$$
12. $$a_j=2\cdot j-5$$
13. $$a_n=n^2 +8n+15$$
14. $$a_k=9\cdot (k+5) +7k-1$$

For the convenience of those who prefer to use $$a_{n}=a+b \cdot n$$ as standard form we have provided answers also in that form.

1. $$5 + 3(n−1) = 2 + 3n$$
2. $$−10 + 3(n−1) = −13 + 3n$$
3. no
4. no
5. $$73.4−21.7(n−1) = 95.1−21.7n$$
6. no
7. $$4 + 0 ·(n−1) = 4 + 0 ·n$$
8. $$-2.72-.1(n-1)=-2.62-.1 n$$
9. no
10. $$-\dfrac{3}{5}+\dfrac{1}{2}(n-1)=-\dfrac{11}{10}+\dfrac{1}{2} n$$
11. $$9 + 5(n − 1) = 4 + 5n$$
12. $$−3 + 2(j − 1) = −5 + 2j$$
13. no
14. $$29 + 16(k − 1) = 13 + 16k$$

## Exercise $$\PageIndex{5}$$

Determine the general $$n$$th term $$a_n$$ of an arithmetic sequence $$\{a_n\}$$ with the data given below.

1. $$d=4$$, and $$a_{8}=57$$
2. $$d=-3$$, and $$a_{99}=-70$$
3. $$a_1=14$$, and $$a_{7}=-16$$
4. $$a_1=-80$$, and $$a_{5}=224$$
5. $$a_{3}=10$$, and $$a_{14}=-23$$
6. $$a_{20}=2$$, and $$a_{60}=32$$
1. $$57 + 4(n − 8) = 29 + 4(n − 1) = 25 + 4n$$
2. $$−70 − 3(n − 99) = 224 − 3(n − 1) = 227 − 3n$$
3. $$14 − 5(n − 1) = 19 − 5n$$
4. $$−80 + 76(n − 1) = −156 + 76n$$
5. $$10 − 3(n − 3) = 16 − 3(n − 1) = 19 − 3n$$
6. $$2+\dfrac{3}{4}(n-20)=-49 / 4+\dfrac{3}{4}(n-1)=-13+\dfrac{3}{4} n$$

## Exercise $$\PageIndex{6}$$

Determine the value of the indicated term of the given arithmetic sequence.

1. if $$a_1=8$$, and $$a_{15}=92$$, find $$a_{19}$$
2. if $$d=-2$$, and $$a_3=31$$, find $$a_{81}$$
3. if $$a_1=0$$, and $$a_{17}=-102$$, find $$a_{73}$$
4. if $$a_{7}=128$$, and $$a_{37}=38$$, find $$a_{26}$$
1. $$116$$
2. $$187$$
3. $$-\dfrac{3621}{8}$$
4. $$71$$

## Exercise $$\PageIndex{7}$$

Determine the sum of the arithmetic sequence.

1. Find the sum $$a_1+\dots +a_{48}$$ for the arithmetic sequence $$a_i=4i+7$$.
2. Find the sum $$\sum_{i=1}^{21}a_i$$ for the arithmetic sequence $$a_n=2-5n$$.
3. Find the sum: $$\sum\limits_{i=1}^{99} (10\cdot i+1)$$
4. Find the sum: $$\sum\limits_{n=1}^{200} (-9-n)$$
5. Find the sum of the first $$100$$ terms of the arithmetic sequence: $$2, 4, 6, 8, 10, 12, \dots$$
6. Find the sum of the first $$83$$ terms of the arithmetic sequence: $$25, 21, 17, 13, 9, 5, \dots$$
7. Find the sum of the first $$75$$ terms of the arithmetic sequence: $$2012, 2002, 1992, 1982, \dots$$
8. Find the sum of the first $$16$$ terms of the arithmetic sequence: $$-11, -6, -1, \dots$$
9. Find the sum of the first $$99$$ terms of the arithmetic sequence: $$-8, -8.2, -8.4, -8.6, -8.8, -9, -9.2, \dots$$
10. Find the sum $$7+8+9+10+\dots+776+777$$
11. Find the sum of the first $$40$$ terms of the arithmetic sequence: $$5, 5, 5, 5, 5, \dots$$
1. $$5, 040$$
2. $$−1, 113$$
3. $$49, 599$$
4. $$−21, 900$$
5. $$10, 100$$
6. $$−11, 537$$
7. $$123, 150$$
8. $$424$$
9. $$−1762.2$$
10. $$302, 232$$
11. $$200$$

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