Question

Write the expression $a_n-a_k$ for the A.P. $a,a+d,a+2d,.....$
Hence, find the common difference of the A.P for which $20$th term is $10$ more than the $18$th term.

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

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

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

For finding the $k$th term of sequence $\left(a_k\right)$ put $n=k$ in eq(1)

$⟹a_k=a+(k-1)d\dots .eq\left(2\right)$

now,

$a_n-a_k=(a+(n-1\left)d\right)-(a+(k-1\left)d\right)$

$a_n-a_k=(nd-d-kd+d)$

$a_n-a_k=(n-k)d\dots .eq\left(3\right)$

put n=20 and k= 18 in eq(3) we get

$a_{20}-a_{18}=(10-5)d$

given that $20$th term is $10$ more than $18$th term $⟹a_{20}-a_{18}=10$

$10=(20-18)d$

$10=\left(2\right)d$

$d=\frac{10}{2}$

$d=5$ common difference for the A.P for which $a_{20}-a_{18}=10$