Question

The sum of the first three numbers in anA.P. is 24 and their product is 384 . Then which of the following hold(s) good?

• A. Sum to $n$ terms can be $2n(n+1)$.
• B. Sum to $n$ terms can be $2n(n+3)$.
• C. Sum to $n$ terms can be $14n-2n^2$.
• D. Sum of the squares of the 3 numbers is 224 .

Answer 1

Answer: (A), (C), (D)

Let the number be $(a-d),a,(a+d)$
$3a=24\Rightarrow a=8$
Also $(8-d)(8)(8+d)=384\Rightarrow 64-d^2=48\Rightarrow d^2=16\Rightarrow d=4$ or -4
Hence the series is

$S_n=\frac{n}{2}[8+(n-1)4]=n(4+2n-2)=n(2n+2)=2n(n+1)$
Also $S_n=\frac{n}{2}[24-(n-1)4]=n(12-2n+2)=n(14-2n)=14n-2n^2$