Question

If $\left|3x-1\right|,3,\left|x-3\right|$ are the first three terms of an arithmetic progression, then the sum of the first five terms can be

• A. $5$
• B. $10$
• C. $20$
• D. $30$

Answer 1

Answer: (A)

Case-I : $x<\frac{1}{3}\Rightarrow -3x+1,3,-x+3$ are in $A.P.$
$\Rightarrow 6=-3x+1-x+3$
$\Rightarrow 4x=-2\Rightarrow x=-\frac{1}{2}$
$\Rightarrow$ terms are $\frac{5}{2},3,\frac{7}{2},4,\frac{9}{2}$
$\Rightarrow$ sum $=\frac{35}{2}$
Case-II: $\frac{1}{3}\le x<3$
$3x-1,3,-x+3$ are in A.P.
$\Rightarrow 6=3x-1-x+3=2x+2\Rightarrow x=2$
$\Rightarrow$ terms are $5,3,1,-1,-3$
$\Rightarrow$ sum $=5$
Case-III: $x\ge 3$
$\Rightarrow$ terms are $3x-1,3,x-3$
$\Rightarrow 6=3x-1+x-3=4x-4$
$\Rightarrow 4x=10\Rightarrow x=\frac{5}{2}$ not possible