Consider two sets A and B, each containing three numbers in A.P.

Question

Consider two sets A and B, each containing three numbers in A.P. Let the sum and the product of the elements of A be 36 and p respectively and the sum and the product of the elements of B be 36 and $q$ respectively. Let d and D be the commo...

Accepted Answer

Let the terms in $A$ be ${a}_{1}-d,{a}_{1},{a}_{1}+d$

and in $B$ be ${a}_{2}-D,{a}_{2},{a}_{2}+D$

Now $3{a}_{1}=36$

$\Rightarrow \, {a}_{1}=12$

and $3{a}_{2}=36$

$\Rightarrow \, {a}_{2}=12$

Now $(12-d)(12)(12+d)=p$

and $(12-D)(12)(12+D)=q$

Also $\frac{p+q}{p-q}=\frac{19}{5}$

$\begin{gathered} \Rightarrow 12q=7p \ \Rightarrow 12(12-D)(12)(12+D)=7(12-d)(12)(12+d) \ \Rightarrow 12(9-d)(12)(15-d)=7(12-d)(12)(12+d) \ \Rightarrow 12(135-{d}^{2}-6d)=7(144-{d}^{2}) \ \Rightarrow d=6,D=9 \ p=6	imes 12	imes 18=1296 \ q=756 \ p-q=540 \end{gathered}$

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