Question

If / the last term, d the common difference and 's' the sum of n terms $(n\ne 1)$ and 'a' the first term of an A.P. be connected by the eqution $8ds={(d+2l)}^2,thend=$

• A. 2a
• B. a
• C. 3a
• D. 4a

Answer 1

The correct option is A 2a

$s=\frac{n(a+l)}{2}.--->1(a->$ first term $l->$last term $n->$total terms)
$l=a+(n-1)d$
$n=\frac{(l-a+d)}{d}--->2$

Using 2 in 1, we have
$s=\frac{(l-a+d)(a+l)}{2d}$

but$s=\frac{(d+2l{)}^2}{2d}$
$\frac{(l-a+d)(a+l)}{2d}=\frac{(d+2l{)}^2}{8d}$
On solving we get
$4ad-4a^2+4l^2+4dl=d^2+4l^2+4dl$
$d^2=4ad-4a^2$
$d^2+4a^2-4ad=0$
$(d-2a{)}^2=0$
$d=2a$