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 which11th term is 5 and 13th term is 79.

Given:

Given A.P. is $a, a + d, a + 2d, ……$

11th term is 5 and 13th term is 79.

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,

$a_{11}=a+(11-1)d$

$5=a+10d$

$a=5-10d$......(i)

$a_{13}=a+(13-1)d$

$79=5-10d+12d$      (From (i))

$79-5=2d$

$2d=74$

$d=37$

Hence, $a_{n}-a_{k}$ is $(n-k)d$ and the common difference is $37$. 

Tutorialspoint

Updated on: 10-Oct-2022

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