Question

If $\left|3 x - 1\right|,3,\left|x - 3\right|$ 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: $\frac{1}{3}\leq x < 3$
$3x-1,3,-x+3$ 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: $x\geq 3$
$\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