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

Answer 1

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

Let the number be $(a-d), a,(a+d)$
$3 a =24 \Rightarrow a =8$
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

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