Question

If the roots of the equation $x^3-9x^2+23x-15=0$ are the first three terms of an A.P, then the sum of first $20$ terms of the A.P. can be

• A. $400$
• B. $280$
• C. $-400$
• D. $-280$

Answer 1

Answer: (A), (D)

The correct options are
A $400$
D $-280$

Let roots are $a-d,a,a+d$
$\Rightarrow (a-d)+a+(a+d)=9\Rightarrow a=3\Rightarrow (a-d)a(a+d)=15\Rightarrow d=\pm 2\Rightarrow \text{Roots are}1,3,5\text{when}d=2\Rightarrow \text{Roots are}5,3,1\text{when}d=-2\text{When,}a=1\text{and}d=2S_{20}=\frac{20}{2}[2\times 1+19\times 2]=400\text{When,}a=5\text{and}d=-2S_{20}=\frac{20}{2}[2\times 5+19\times (-2\left)\right]=-280$