Question

The numbers of terms in the A. P $a,b,c,........,x$ is

• A. $\frac{x+b+a}{c-a}$
• B. $\frac{x+b-2a}{c-b}$
• C. $\frac{x+b+2a}{c-b}$
• D. $\frac{x-b+a}{c-b}$

Answer 1

The correct option is B

$\frac{x+b-2a}{c-b}$

Find the number of terms in the given Arithmetic Progression

Given AP, $a,b,c,........,x$

If a series of numbers are in arithmetic progression, the difference between consecutive terms is constant.

The common difference $d$ is given by

$d=b-a=c-b$

The $n^{th}$ term of an arithmetic progression is given as

${\mathit{t}}_{\mathbf{n}}\mathbf{=}\mathit{a}\mathbf{+}\left(n-1\right)\mathit{d}$ where $a$is the first term, $d$is the common difference

Substitute all the given values

$a+\left(n-1\right)\left(c-b\right)=x$

$\Rightarrow$ $\left(n-1\right)\left(c-b\right)=x-a$

$\Rightarrow$ $\left(n-1\right)=\frac{x-a}{c-b}$

$\Rightarrow$ $n=\frac{x-a}{c-b}+1$

$\Rightarrow$ $n=\frac{x-a+c-b}{c-b}$

$\Rightarrow$ $n=\frac{x-a+b-a}{c-b}$

$\Rightarrow$ $n=\frac{x+b-2a}{c-b}$

there are $\frac{x+b-2a}{c-b}$ terms in the given arithmetic progression so,

Hence, option $\left(B\right)$ is the correct answer.