Question

The sum of three decreasing numbers in A.P. is $27$. If $-1,-1,3$ are added to them respectively, the resulting series is in G.P. The numbers are

• A. $5,9,13$
• B. $15,9,3$
• C. $13,9,5$
• D. $17,9,1$

Answer 1

The correct option is D

$17,9,1$

Explanation of the correct option:

Let $\mathrm{a}+\mathrm{d},\mathrm{a},\mathrm{a}-\mathrm{d}$ be three terms of an A.P.

Sum$=\mathrm{a}+\mathrm{d}+\mathrm{a}+\mathrm{a}-\mathrm{d}$

Given : The sum of three decreasing numbers in A.P. is $27$.

Thus, $3\mathrm{a}=27$

$\Rightarrow$ $\mathrm{a}=9$

Add $-1,-1,3$ to the numbers, then we have

$9+\mathrm{d}-1,9-1,9-\mathrm{d}+3$

Now $8+\mathrm{d},8,12-\mathrm{d}$ are in G.P.

Apply condition of G.P.

$8^2=(8+\mathrm{d})(12-\mathrm{d})$

$\Rightarrow$ $64=96+12\mathrm{d}-8\mathrm{d}-{\mathrm{d}}^2$

$\Rightarrow$ ${\mathrm{d}}^2-4\mathrm{d}-32=0$

$\Rightarrow$$\left(\mathrm{d}-8\right)\left(\mathrm{d}+4\right)=0$

Thus, $\mathrm{d}=8$ or $\mathrm{d}=-4$

When $\mathrm{d}=8$, the numbers are $17,9,1$.

When $\mathrm{d}=-4$, the numbers are $5,9,13$ (not possible since A.P. is decreasing.)

Hence option D is the correct option.