Question

In a HP the $7^{th}$ term is $\frac{1}{7}$ and ${14}^{th}$ term is $1$. Find the third term .

• A. $10.45$
• B. $12.45$
• C. $14.65$
• D. $15.25$

Answer 1

The correct option is A $10.45$

$7^{th}$ term of HP = $\frac{1}{7}$, then the reciprocal of $7^{th}$ term is AP $=7$.
$t_7=7$
But we know that
$t_n=a+(n-1)d$
$t_7=a+6d$
$a+6d=7$---- (1)
According to the problem, ${14}^{th}$ term of HP is $1$.
Then the reciprocal of AP is $1$.
Hence, ${14}^{th}$ term of AP is $1$
$t_{14}=1$
$t_{14}=a+(n-1)d$
$a+13d=1$---- (2)
Equating equation 1 and 2,
$-7d=6$
$d=\frac{6}{-7}$
Substitute the value of d in equation (1),
$a+6\left(\frac{6}{-7}\right)=7$
$-7a+36=-49$
$-7a=-49-36$
$a=12.14$
We know the general term of AP given by $t_n=a+(n-1)d$
Third term of AP, $t_3=12.14+(3-1)\frac{6}{-7}$
$t_3=12.14-\frac{12}{7}$
$t_3=10.45$