Question

If the sum of three numbers of a arithmetic sequence is 15 and the sum of their squares is 83, then the numbers are.

• A. 4, 5, 6
• B. 3, 5, 7
• C. 1, 5, 9
• D. 2, 5, 8

Answer 1

Answer: (B)

3, 5, 7

Answer 2

Let three numbers are $a - d , a , a + d$.

We get $a - d + a + a + d = 15$ $\Rightarrow$ $a = 5$

And $( a - d ) ^ { 2 } + a ^ { 2 } + ( a + d ) ^ { 2 } = 83$

$\Rightarrow$ $a ^ { 2 } + d ^ { 2 } - 2 a d + a ^ { 2 } + a ^ { 2 } + d ^ { 2 } + 2 a d = 83$

$\Rightarrow$ $2 \left( a ^ { 2 } + d ^ { 2 } \right) + a ^ { 2 } = 83$

Putting $a = 5$

$\Rightarrow$ $2 \left( 25 + d ^ { 2 } \right) + 25 = 83$ $\Rightarrow$ $2 d ^ { 2 } = 8$ $\Rightarrow$ $d = 2$

Thus numbers are 3, 5, 7.

Trick : Since $3 + 5 + 7 = 15$ and $3 ^ { 2 } + 5 ^ { 2 } + 7 ^ { 2 } = 83$.