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  4. If x, |x + 1| and |x - 1| are the first three terms of an arithmetic progression, then its sum upto 20 terms is

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Q. 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

NTA AbhyasNTA Abhyas 2020Sequences and Series

Solution:

$2\left|x + 1\right|=x+\left|x - 1\right|$
(i) Case- $Ι$ :-
$x\leq -1$
$2\left(- x - 1\right)=x+\left(- x + 1\right)\Rightarrow -2x-2=1$
$\Rightarrow x=-\frac{3}{2}$
Then the A.P. is $ \rightarrow -\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- $ΙΙ$ :-
$-1\leq x < 1$
$2\left(x + 1\right)=x+\left(- x + 1\right)\Rightarrow 2x+2=1\Rightarrow x=-\frac{1}{2}$
Then the A.P. is $ \rightarrow -\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- $ΙΙΙ$ :-
$x\geq 1$
image
(not possible)
Hence, the required sum is $350$ or $180$