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 which20th term is 10 more than the 18th term.

Given:

Given A.P. is $a, a + d, a + 2d, ……$

20th term is 10 more than the 18th term.

To do:

We have to find $a_{n} - a_{k}$ and the common difference of the A.P.

Solution:

$a_1=a, a_2=a+d, a_3=a+2d$ and $d=a_2-a_1=a+d-(a)=a+d-a=d$

nth term of the A.P. $a_n=a+(n-1)d$

kth term of the A.P. $a_k=a+(k-1)d$

$a_n-a_k=a+(n-1)d-[a+(k-1)d]$

$=a+nd-d-a-kd+d$

$=(n-k)d$

According to the question,

20th term is 10 more than the 18th term.

$a_{20}=a+(20-1)d$

$=a+19d$

$a_{18}=a+(18-1)d$

$=a+17d$

This implies,

$a+19d=(a+17d)+10$

$a+19d-a-17d=10$

$2d=10$

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

$d=5$

Hence, $a_{n}-a_{k}$ is $(n-k)d$ and the common difference is $5$.   

Tutorialspoint

Updated on: 10-Oct-2022

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