Question

Find $a_{30}-a_{20}$ for the A.P
$a,a+d,a+2d,a+3d,........$

Answer 1

Given A.P is $a,a+d,a+2d,a+3d,....$

hence the common difference is given by $d=a_{n+1}-a_n$

by putting n=1 in above equation

$d=a_2-a_1=(a+d)-\left(d\right)$

$d=d$

first term of this A.P is

$a_1=a$

the nth term of this A.P is given by

$a_n=a_1+(n-1)d$

$⟹a_n=a+(n-1)d\dots .eq\left(1\right)$

1) For finding the 30th term of sequence $\left(a_{30}\right)$ put n=30 in eq(1)

$⟹a_{30}=a+(30-1)d$

$⟹a_{30}=a+29d$

2) For finding the 20th term of sequence $\left(a_{20}\right)$put n=20 in eq(1)

$⟹a_{20}=a+(20-1)$

$⟹a_{20}=a+19d$

now,

$a_{30}-a_{20}=(a+29d)-(a+19d)$

$a_{30}-a_{20}=(29d-19d)$

$a_{30}-a_2=-10d$