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

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 kth 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,

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

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

$a_n-a_k=(n-k)d$

given 11th term of A.P is 5

$a_{11}=5=a+(11-1)d$

$5=a+10d......eq\left(3\right)$

given 13th term of A.P is 79

$a_{13}=79=a+(13-1)d$

$79=a+12d......eq\left(4\right)$

substract eq(3) from eq(4) we get

$79-5=a-a+(12-10)d$

$74=2d$

$⟹d=37$ common difference