Question

Write down the sequence whose $n^{th}$ term is
(i) $\frac{2^n}{n}$ (ii) $\frac{3+{(-1)}^n}{3^n}$. How many terms of the sequences are are less than zero?

Answer 1

Let $t_n=\frac{2^n}{n}$

Put n = 1, 2, 3, 4, ........ we get

$t_1=\frac{2}{1}=2$

$t_2=\frac{4}{2}=2$

$t_3=\frac{8}{3}$

$t_4=\frac{16}{4}=4$

So the sequence is $2,2,\frac{8}{3},4,.......$

(ii) Let $t_n=\frac{3+{(-1)}^n}{3^n}$

Put n = 1, 2, 3, 4......

$t_1=\frac{3+(-1)}{3}=\frac{2}{3}$

$t_2=\frac{3+(-1{)}^2}{(3{)}^2}=\frac{4}{9}$

$t_3=\frac{3+(-1{)}^3}{(3{)}^3}=\frac{2}{27}$

$t_4=\frac{3+(-1{)}^4}{(3{)}^4}=\frac{4}{81}$

So the sequence is $\frac{2}{3},\frac{4}{9},\frac{2}{27},\frac{4}{81},....$