Question

Write the first five terms in each of the following sequences :

(i) $a_1=1,a_n=a_{n-1}+2,n>1$

(ii) $a_1=1=a_2,a_n=a_{n-1}+a_n-2,n>2$

(iii) $a_1=a_2=2,a_n=a_{n-1}-1,n>2$

Answer 1

(i) $a_1=1,a_n=a_{n-1}+2,n\ge 2$

$a_2=a_{2-1}+2=a_1+2=3$ $[\because a_1=1]$

$a_3=a_{3-1}+2=a_2+2=5$ $[\because a_2=3]$

$a_4=a_{4-1}+2=a_3+2=7$ $[\because a_3=5]$

$a_5=a_{5-1}+2=a_4+2=9$ $\because a_4=7$

$\therefore$ The first 5 terms of series are 1, 3, 5, 7, 9.

(ii) $a_1=1$

$a_n=a_{n-1}+a_{n-2},n>2,a=1$

$\Rightarrow a_3=a_{3-1}+a_{3-2}$

$=a_2+a_1=1+1=2$

$\Rightarrow a_4=a_{4-1}+a_{4-2}$

$=a_3+a_2=2+1=3$

$\Rightarrow a_5=a_{5-1}+a_{5-2}$

$=a_4+a_3=5$

$\therefore$ The given sequence is 1, 1, 2, 3, 5.

(iii) $a_1-a_2=2$

$a_n=a_{n-1}-1n>2$

$\Rightarrow a_3=a_{3-1}-1$

$=a_2-1=2-1=1$

$\Rightarrow a_4=a_{4-1}-1$

$=a_3-1=1-1=0$

$\Rightarrow a_5=a_{5-1}-1$

$=0-1=-1$

$\therefore$ The first 5 terms of the sequence are 2, 2, 1, 0, -1.