Question

If the $5$th term of an A.P is eight times the first term and $8$th term exceeds twice the $4$th term by $3$, then the common difference is

• A. $7$
• B. $5$
• C. $6$
• D. $3$

Answer 1

The correct option is A $7$

$n^{th}$ term of an AP is given as,

$t_n=a+(n-1)d$

Where $a$ and $d$ are first term and common difference respectively.

According to the given condition,

Thus,

$t_5=a+4d$

$t_1=a$

$t_8=a+7d$

$t_4=a+3d$
Given, $t_5=8t_1$
$⟹a+4d=8a⟹4d=7a⟹d=\frac{7a}{4}$

Also given, $t_8$ $-2t_4=3$

$⟹a+7d-2(a+3d)=3⟹d-a=3$

Substituting $d=\frac{7a}{4}$ in the above equation,

$\frac{7a}{4}-a=3$

$\frac{3a}{4}=3$

we get $a=4$ and hence, $d=7$.