Question

If $x,\left|x+1\right|$ and $\left|x-1\right|$ are the first three terms of an arithmetic progression, then its sum upto $20$ terms is

• A. $90$ or $175$
• B. $180$ or $350$
• C. $360$ or $700$
• D. $720$ or $1400$

Answer 1

Answer: (B)

$2\left|x+1\right|=x+\left|x-1\right|$
(i) Case- $I$ :-
$x\le -1$
$2\left(-x-1\right)=x+\left(-x+1\right)\Rightarrow -2x-2=1$
$\Rightarrow x=-\frac{3}{2}$
Then the A.P. is $\to -\frac{3}{2},\frac{1}{2},\frac{5}{2}......$
$S_{20}=\frac{20}{2}\left[-3+19\times 2\right]=\frac{20}{2}\times 35=350$
(ii) Case- $II$ :-
$-1\le x<1$
$2\left(x+1\right)=x+\left(-x+1\right)\Rightarrow 2x+2=1\Rightarrow x=-\frac{1}{2}$
Then the A.P. is $\to -\frac{1}{2},\frac{1}{2},\frac{3}{2}......$
$S_{20}=\frac{20}{2}\left[-1+19\times 1\right]=10\times 18=180$
(ii) Case- $III$ :-
$x\ge 1$

(not possible)
Hence, the required sum is $350$ or $180$