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
(i) 11th term is 5 and 13th term is 79.
(ii) $a_{10}-a_5=200$
(iii) 20th term is 10 more than the 18th term.

Answer 1

Solving we have,

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

$a_n$ −$a_k=(n-1)d-(k-1)d$

$a_n$ −$a_k=d[n-k]$

$\left(1\right)$ Given, 11 th term is 5 and 13 th term is 79

Substituting the values, we get

$a_{13}$ − $a_{11}=d[13-11]\Rightarrow 79-5=d\left[2\right]\Rightarrow d=37$

$\left(2\right)$

$a_{10}$ -$a_5=200\Rightarrow a+9d-(a+4d)=200\Rightarrow 5d=200\Rightarrow d=40$

$\left(3\right)$
$a_{20}$ -$a_{18}=10$
$10=d(20-18)$
$d=5$