Question

If $\alpha =x_{1}+x_{3}+\ldots x_{199}$ and $\beta =x_{2}+x_{4}+\ldots x_{200}$ , such that $x_{1},x_{2},x_{3},...............x_{200}$ are in arithmetic progression, then their common difference is

• A. $\beta -\alpha $
• B. $\frac{\beta - \alpha }{2}$
• C. $\frac{\alpha - \beta }{10}$
• D. $\frac{\beta - \alpha }{100}$

Answer 1

Answer: (D)

In the $A.P.$ $x_{1}, \, x_{2}, \, x_{3}, \, \ldots \ldots \ldots .x_{200}$ let the common difference be $d$ .
$\therefore \alpha =x_{1}+x_{3}+ \, \ldots \ldots \ldots .+x_{199}=\frac{100}{2}\left(x_{1} + x_{199}\right)$ $=50\left(2 x_{1} + 198 d\right)$
$\beta =x_{2}+x_{4}+ \, \ldots \ldots \ldots x_{200}=\frac{100}{2}\left(x_{2} + x_{200}\right) \, \, $
= $50\left(x_{1} + d + x_{1} + 199 d\right)$
$=50\left(2 x_{1} + 200 d\right)$
$\beta -\alpha =100d \, $
$d=\frac{\beta - \alpha }{100}$