Question

Find the next five terms of each of the following sequences given by:$a_1=1,a_n=a_{n-1}+2,n\ge 2$

Answer 1

The $n^{th}$ term of the sequence is

$a_n=a_{n-1}+2forn\ge 2.............eq\left(1\right)$

and

$a_n=1forn=1$

$⟹a_1=1.......eq\left(2\right)$

1) To find the second term of sequence $\left(a_2\right)$ put n=2 in eq(1)

$⟹a_2=a_{2-1}+2$

$⟹a_2=a_1+2$

put value of $a_1$ from eq(2)

$⟹a_2=1+2$

$⟹a_2=3......eq\left(3\right)$

2)To find the third term of seqence $\left(a_3\right)$ put n=3 in eq(1)

$⟹a_3=a_{3-1}+2$

$⟹a_3=a_2+2$

put value of $a_2$ from eq(3)

$⟹a_3=3+2$

$⟹a_3=5.......eq\left(4\right)$

3)To find the fourth term of sequence $\left(a_4\right)$ put n=4 in eq(1)

$⟹a_4=a_{4-1}+2$

$⟹a_4=a_3+2$

put value of $a_3$ from eq(4)

$⟹a_4=5+2$

$⟹a_4=7....eq\left(5\right)$

4)To find the fifth term of sequence $\left(a_5\right)$ put n=5 in eq(1)

$⟹a_5=a_{5-1}+2$

$⟹a_5=a_4+2$

put value of $a_4$ from eq(5)

$⟹a_5=7+2$

$⟹a_5=9....eq\left(6\right)$

5)To find the sixth term of sequence $\left(a_6\right)$ put n=6 in eq(1)

$⟹a_6=a_{6-1}+2$

$⟹a_6=a_5+2$

put value of $a_5$ from eq(6)

$⟹a_6=9+2$

$⟹a_6=11....eq\left(7\right)$