Question

The sum of the first four terms of an A.P is $56$. The sum of the last four terms is $112$. If its first term is $11$ , the number of terms is

• A. $10$
• B. $11$
• C. $12$
• D. None of these

Answer 1

The correct option is B

$11$

Finding the no of terms:

Let $a$ be the first term of the arithmetic progression and $d$ be the common difference.

The first four terms of the arithmetic progression are $a,a+d,a+2d,a+3d$

$a=11$ …(given)

The sum of the first four terms is,

$a+a+d+a+2d+a+3d=56$

$\Rightarrow$ $4a+6d=56$

$\Rightarrow$ $44+6d=56$

$\Rightarrow$ $6d=12$

$\Rightarrow$ $d=2$

The last four terms of the arithmetic progression can be written as

$a+(n-4)d,a+(n-3)d,a+(n-2)d,a+(n-1)d$

The sum of the last four terms is $112$

$\Rightarrow$ $a+\left(n-4\right)d+a+\left(n-3\right)d+a+\left(n-2\right)d+a+\left(n-1\right)d=112$

$\Rightarrow$ $4a+4nd-10d=112$

Substitute values of $a,d$

$\Rightarrow$ $4\times 11+4n\times 2-10\times 2=112$

$\Rightarrow$ $8n=88$

$\Rightarrow$ $n=11$

Hence, there are $11$ terms in the arithmetic progression.

Hence, option (B) is the correct answer.