# 12.2E: Exercises

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## Practice Makes Perfect

##### Exercise $$\PageIndex{25}$$ Write the First Few Terms of a Sequence

In the following exercises, write the first five terms of the sequence whose general term is given.

1. $$a_{n}=2 n-7$$
2. $$a_{n}=5 n-1$$
3. $$a_{n}=3 n+1$$
4. $$a_{n}=4 n+2$$
5. $$a_{n}=2^{n}+3$$
6. $$a_{n}=3^{n}-1$$
7. $$a_{n}=3^{n}-2 n$$
8. $$a_{n}=2^{n}-3 n$$
9. $$a_{n}=\frac{2^{n}}{n^{2}}$$
10. $$a_{n}=\frac{3^{n}}{n^{3}}$$
11. $$a_{n}=\frac{4 n-2}{2^{n}}$$
12. $$a_{n}=\frac{3 n+3}{3^{n}}$$
13. $$a_{n}=(-1)^{n} \cdot 2 n$$
14. $$a_{n}=(-1)^{n} \cdot 3 n$$
15. $$a_{n}=(-1)^{n+1} n^{2}$$
16. $$a_{n}=(-1)^{n+1} n^{4}$$
17. $$a_{n}=\frac{(-1)^{n+1}}{n^{2}}$$
18. $$a_{n}=\frac{(-1)^{n+1}}{2 n}$$

1. $$-5,-3,-1,1,3$$

3. $$4,7,10,13,16$$

5. $$5,7,11,19,35$$

7. $$1,5,21,73,233$$

9. $$2,1, \frac{8}{9}, 1, \frac{32}{25}$$

11. $$1, \frac{3}{2}, \frac{5}{4}, \frac{7}{8}, \frac{9}{16}$$

13. $$-2,4,-6,8,-10$$

15. $$1,-4,9,-16,25$$

17. $$1,-\frac{1}{4}, \frac{1}{9},-\frac{1}{16}, \frac{1}{25}$$

##### Exercise $$\PageIndex{26}$$ Find a Formula for the General Term ($$n$$th Term) of a Sequence

In the following exercises, find a general term for the sequence whose first five terms are shown.

1. $$8,16,24,32,40, \dots$$
2. $$7,14,21,28,35, \ldots$$
3. $$6,7,8,9,10, \dots$$
4. $$-3,-2,-1,0,1, \dots$$
5. $$e^{3}, e^{4}, e^{5}, e^{6}, e^{7}, \ldots$$
6. $$\frac{1}{e^{2}}, \frac{1}{e}, 1, e, e^{2}, \ldots$$
7. $$-5,10,-15,20,-25, \dots$$
8. $$-6,11,-16,21,-26, \dots$$
9. $$-1,8,-27,64,-125, \dots$$
10. $$2,-5,10,-17,26, \dots$$
11. $$-2,4,-6,8,-10, \dots$$
12. $$1,-3,5,-7,9, \dots$$
13. $$\frac{1}{4}, \frac{1}{16}, \frac{1}{64}, \frac{1}{256}, \frac{1}{1,024}, \dots$$
14. $$\frac{1}{1}, \frac{1}{8}, \frac{1}{27}, \frac{1}{64}, \frac{1}{125}, \dots$$
15. $$-\frac{1}{2},-\frac{2}{3},-\frac{3}{4},-\frac{4}{5},-\frac{5}{6}, \dots$$
16. $$-2,-\frac{3}{2},-\frac{4}{3},-\frac{5}{4},-\frac{6}{5}, \dots$$
17. $$-\frac{5}{2},-\frac{5}{4},-\frac{5}{8},-\frac{5}{16},-\frac{5}{32}, \dots$$
18. $$4, \frac{1}{2}, \frac{4}{27}, \frac{4}{64}, \frac{4}{125}, \dots$$

1. $$a_{n}=8 n$$

3. $$a_{n}=n+5$$

5. $$a_{n}=e^{n+2}$$

7. $$a_{n}=(-1)^{n} 5 n$$

9. $$a_{n}=(-1)^{n} n^{3}$$

11. $$a_{n}=(-1)^{n} 2 n$$

13. $$a_{n}=\frac{1}{4^{n}}$$

15. $$a_{n}=-\frac{n}{n+1}$$

17. $$-\frac{5}{2^{n}}$$

##### Exercise $$\PageIndex{27}$$ Use Factorial Notation

In the following exercises, using factorial notation, write the first five terms of the sequence whose general term is given.

1. $$a_{n}=\frac{4}{n !}$$
2. $$a_{n}=\frac{5}{n !}$$
3. $$a_{n}=3 n !$$
4. $$a_{n}=2 n !$$
5. $$a_{n}=(2 n) !$$
6. $$a_{n}=(3 n) !$$
7. $$a_{n}=\frac{(n-1) !}{(n) !}$$
8. $$a_{n}=\frac{n !}{(n+1) !}$$
9. $$a_{n}=\frac{n !}{n^{2}}$$
10. $$a_{n}=\frac{n^{2}}{n !}$$
11. $$a_{n}=\frac{(n+1) !}{n^{2}}$$
12. $$a_{n}=\frac{(n+1) !}{2 n}$$

1. $$4,2, \frac{2}{3}, \frac{1}{6}, \frac{1}{30}$$

3. $$3,6,18,72,360$$

5. $$2,24,720,40320,3628800$$

7. $$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \frac{1}{5}$$

9. $$1, \frac{1}{2}, \frac{2}{3}, \frac{3}{2}, \frac{24}{5}$$

11. $$2, \frac{3}{2}, \frac{8}{3}, \frac{15}{2}, \frac{144}{5}$$

##### Exercise $$\PageIndex{28}$$ Find the Partial Sum

In the following exercises, expand the partial sum and find its value.

1. $$\sum_{i=1}^{5} i^{2}$$
2. $$\sum_{i=1}^{5} i^{3}$$
3. $$\sum_{i=1}^{6}(2 i+3)$$
4. $$\sum_{i=1}^{6}(3 i-2)$$
5. $$\sum_{i=1}^{4} 2^{i}$$
6. $$\sum_{i=1}^{4} 3^{i}$$
7. $$\sum_{k=0}^{3} \frac{4}{k !}$$
8. $$\sum_{k=0}^{4}-\frac{1}{k !}$$
9. $$\sum_{k=1}^{5} k(k+1)$$
10. $$\sum_{k=1}^{5} k(2 k-3)$$
11. $$\sum_{n=1}^{5} \frac{n}{n+1}$$
12. $$\sum_{n=1}^{4} \frac{n}{n+2}$$

1. $$1+4+9+16+25=55$$

3. $$5+7+9+11+13+15=60$$

5. $$2+4+8+16=30$$

7. $$\frac{4}{1}+\frac{4}{1}+\frac{4}{2}+\frac{4}{6}+\frac{32}{3}=10 \frac{2}{3}$$

9. $$2+6+12+20+30=70$$

11. $$\frac{1}{2}+\frac{2}{3}+\frac{3}{4}+\frac{4}{5}+\frac{5}{6}=\frac{71}{20}$$

##### Exercise $$\PageIndex{29}$$ Use Summation Notation to Write a Sum

In the following exercises, write each sum using summation notation.

1. $$\frac{1}{3}+\frac{1}{9}+\frac{1}{27}+\frac{1}{81}+\frac{1}{243}$$
2. $$\frac{1}{4}+\frac{1}{16}+\frac{1}{64}+\frac{1}{256}$$
3. $$1+\frac{1}{8}+\frac{1}{27}+\frac{1}{64}+\frac{1}{125}$$
4. $$\frac{1}{5}+\frac{1}{25}+\frac{1}{125}+\frac{1}{625}$$
5. $$2+1+\frac{2}{3}+\frac{1}{2}+\frac{2}{5}$$
6. $$3+\frac{3}{2}+1+\frac{3}{4}+\frac{3}{5}+\frac{1}{2}$$
7. $$3-6+9-12+15$$
8. $$-5+10-15+20-25$$
9. $$-2+4-6+8-10+\ldots+20$$
10. $$1-3+5-7+9+\ldots+21$$
11. $$14+16+18+20+22+24+26$$
12. $$9+11+13+15+17+19+21$$

1. $$\sum_{n=1}^{5} \frac{1}{3^{n}}$$

3. $$\sum_{n=1}^{5} \frac{1}{n^{3}}$$

5. $$\sum_{n=1}^{5} \frac{2}{n}$$

7. $$\sum_{n=1}^{5}(-1)^{n+1} 3 n$$

9. $$\sum_{n=1}^{10}(-1)^{n} 2 n$$

11. $$\sum_{n=1}^{7}(2 n+12)$$

##### Exercise $$\PageIndex{30}$$ Writing Exercises
1. In your own words, explain how to write the terms of a sequence when you know the formula. Show an example to illustrate your explanation.
2. Which terms of the sequence are negative when the $$n^{th}$$ term of the sequence is $$a_{n}=(-1)^{n}(n+2)$$?
3. In your own words, explain what is meant by $$n!$$ Show some examples to illustrate your explanation.
4. Explain what each part of the notation $$\sum_{k=1}^{12} 2 k$$ means.

## Self Check

a. After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.