CAT 2021 Slot 2 QA Question 17: For a sequence of real numbers $x_1, x_2, \ldots, x_n$, if $x_1 - x_2 + x_3 - \ldots + (-1)^{n+1} x_n = n^2 + 2n$ for all natural numbers $n$, then the sum $x_{49}+x_{50}$ equals

Question

For a sequence of real numbers $x_1, x_2, \ldots, x_n$, if $x_1 - x_2 + x_3 - \ldots + (-1)^{n+1} x_n = n^2 + 2n$ for all natural numbers $n$, then the sum $x_{49}+x_{50}$ equals

Accepted Answer

Let
$S_n=x_1-x_2+x_3-\ldots+(-1)^{n+1}x_n=n^2+2n$

Then each term can be found by subtracting consecutive sums.

$x_{49}=S_{49}-S_{48}$
$=(49^2+2 	imes 49)-(48^2+2 	imes 48)$
$=2499-2400=99$

For $n = 50$, the 50th term enters with a negative sign.
$S_{50}-S_{49}=-x_{50}$
$-x_{50}=(50^2+2 	imes 5