Sequences and Series
Exercise 8.1 Solutions
Q1
1. Write the first five terms of the sequence: $a_n = n(n+2)$
n = 1
$a_1 = 1(1+2) = 1(3) = 3$
n = 2
$a_2 = 2(2+2) = 2(4) = 8$
n = 3
$a_3 = 3(3+2) = 3(5) = 15$
n = 4
$a_4 = 4(4+2) = 4(6) = 24$
n = 5
$a_5 = 5(5+2) = 5(7) = 35$
Terms: 3, 8, 15, 24, 35
Q2
2. Write the first five terms: $a_n = \frac{n}{n+1}$
n = 1
$a_1 = \frac{1}{1+1} = \frac{1}{2}$
n = 2
$a_2 = \frac{2}{2+1} = \frac{2}{3}$
n = 3
$a_3 = \frac{3}{3+1} = \frac{3}{4}$
n = 4
$a_4 = \frac{4}{4+1} = \frac{4}{5}$
n = 5
$a_5 = \frac{5}{5+1} = \frac{5}{6}$
Terms: $\frac{1}{2}, \frac{2}{3}, \frac{3}{4}, \frac{4}{5}, \frac{5}{6}$
Q3
3. Write the first five terms: $a_n = 2^n$
Calc
$a_1 = 2^1 = 2$
$a_2 = 2^2 = 4$
$a_3 = 2^3 = 8$
$a_4 = 2^4 = 16$
$a_5 = 2^5 = 32$
$a_2 = 2^2 = 4$
$a_3 = 2^3 = 8$
$a_4 = 2^4 = 16$
$a_5 = 2^5 = 32$
Terms: 2, 4, 8, 16, 32
Q4
4. Write the first five terms: $a_n = \frac{2n-3}{6}$
n = 1
$a_1 = \frac{2(1)-3}{6} = \frac{-1}{6}$
n = 2
$a_2 = \frac{2(2)-3}{6} = \frac{1}{6}$
n = 3
$a_3 = \frac{2(3)-3}{6} = \frac{3}{6} = \frac{1}{2}$
n = 4
$a_4 = \frac{2(4)-3}{6} = \frac{5}{6}$
n = 5
$a_5 = \frac{2(5)-3}{6} = \frac{7}{6}$
Terms: $-\frac{1}{6}, \frac{1}{6}, \frac{1}{2}, \frac{5}{6}, \frac{7}{6}$
Q5
5. Write the first five terms: $a_n = (-1)^{n-1} 5^{n+1}$
n = 1
$a_1 = (-1)^0 \cdot 5^2 = 1 \cdot 25 = 25$
n = 2
$a_2 = (-1)^1 \cdot 5^3 = -1 \cdot 125 = -125$
n = 3
$a_3 = (-1)^2 \cdot 5^4 = 1 \cdot 625 = 625$
n = 4
$a_4 = (-1)^3 \cdot 5^5 = -1 \cdot 3125 = -3125$
n = 5
$a_5 = (-1)^4 \cdot 5^6 = 1 \cdot 15625 = 15625$
Terms: 25, -125, 625, -3125, 15625
Q6
6. Write the first five terms: $a_n = n \frac{n^2+5}{4}$
n = 1
$a_1 = 1 \cdot \frac{1+5}{4} = \frac{6}{4} = \frac{3}{2}$
n = 2
$a_2 = 2 \cdot \frac{4+5}{4} = 2 \cdot \frac{9}{4} = \frac{9}{2}$
n = 3
$a_3 = 3 \cdot \frac{9+5}{4} = 3 \cdot \frac{14}{4} = \frac{21}{2}$
n = 4
$a_4 = 4 \cdot \frac{16+5}{4} = 21$
n = 5
$a_5 = 5 \cdot \frac{25+5}{4} = 5 \cdot \frac{30}{4} = \frac{75}{2}$
Terms: $\frac{3}{2}, \frac{9}{2}, \frac{21}{2}, 21, \frac{75}{2}$
Find Terms
7. Find $a_{17}$ and $a_{24}$ for $a_n = 4n – 3$.
8. Find $a_7$ for $a_n = \frac{n^2}{2^n}$.
9. Find $a_9$ for $a_n = (-1)^{n-1}n^3$.
10. Find $a_{20}$ for $a_n = \frac{n(n-2)}{n+3}$.
a_17
$a_{17} = 4(17) – 3 = 68 – 3 = 65$
a_24
$a_{24} = 4(24) – 3 = 96 – 3 = 93$
65, 93
8. Find $a_7$ for $a_n = \frac{n^2}{2^n}$.
a_7
$a_7 = \frac{7^2}{2^7} = \frac{49}{128}$
49/128
9. Find $a_9$ for $a_n = (-1)^{n-1}n^3$.
a_9
$a_9 = (-1)^{9-1}(9)^3 = (-1)^8(729) = 1 \cdot 729 = 729$
729
10. Find $a_{20}$ for $a_n = \frac{n(n-2)}{n+3}$.
a_20
$a_{20} = \frac{20(20-2)}{20+3} = \frac{20(18)}{23} = \frac{360}{23}$
360/23
Recursive
11. $a_1 = 3, a_n = 3a_{n-1} + 2$ for $n > 1$. Write first 5 terms.
a_1
Given 3
a_2
$3a_1 + 2 = 3(3) + 2 = 11$
a_3
$3a_2 + 2 = 3(11) + 2 = 35$
a_4
$3a_3 + 2 = 3(35) + 2 = 107$
a_5
$3a_4 + 2 = 3(107) + 2 = 323$
3, 11, 35, 107, 323
Recursive
12. $a_1 = -1, a_n = \frac{a_{n-1}}{n}$ for $n \ge 2$. Write first 5 terms.
a_1
-1
a_2
$\frac{a_1}{2} = \frac{-1}{2}$
a_3
$\frac{a_2}{3} = \frac{-1/2}{3} = \frac{-1}{6}$
a_4
$\frac{a_3}{4} = \frac{-1/6}{4} = \frac{-1}{24}$
a_5
$\frac{a_4}{5} = \frac{-1/24}{5} = \frac{-1}{120}$
-1, -1/2, -1/6, -1/24, -1/120
Recursive
13. $a_1 = a_2 = 2, a_n = a_{n-1} – 1$ for $n > 2$. Write first 5 terms.
a_1, a_2
2, 2
a_3
$a_2 – 1 = 2 – 1 = 1$
a_4
$a_3 – 1 = 1 – 1 = 0$
a_5
$a_4 – 1 = 0 – 1 = -1$
2, 2, 1, 0, -1
Fibonacci
14. Fibonacci: $1 = a_1 = a_2$ and $a_n = a_{n-1} + a_{n-2}$. Find $\frac{a_{n+1}}{a_n}$ for $n=1,2,3,4,5$.
Terms
$a_1=1, a_2=1$
$a_3 = 1+1=2$
$a_4 = 2+1=3$
$a_5 = 3+2=5$
$a_6 = 5+3=8$
$a_3 = 1+1=2$
$a_4 = 2+1=3$
$a_5 = 3+2=5$
$a_6 = 5+3=8$
n = 1
$\frac{a_2}{a_1} = \frac{1}{1} = 1$
n = 2
$\frac{a_3}{a_2} = \frac{2}{1} = 2$
n = 3
$\frac{a_4}{a_3} = \frac{3}{2}$
n = 4
$\frac{a_5}{a_4} = \frac{5}{3}$
n = 5
$\frac{a_6}{a_5} = \frac{8}{5}$
1, 2, 3/2, 5/3, 8/5