If the sum of three numbers which are in A.P is 15 and the sum of the

1.	If the sum of three numbers which are in A.P is 15 and the sum of the squares of the extremes is 58, then those numbers are A) 3, 6, 9 B) 3, 5, 7C) 2, 5, 8 D) 2, 6, 7
Answer» Correct option is (B) 3, 5, 7 Let a - d, a, a+d are required three numbers in A.P. Given that their sum is 15. \(\therefore\) (a - d) + a + (a+d) = 15 \(\Rightarrow3a=15\) \(\Rightarrow a=\frac{15}3=5\) Also given that the sum of the squares of the extremes is 58. \(\therefore(a-d)^2+(a+d)^2=58\) \(\Rightarrow(5-d)^2+(5+d)^2=58\) \((\because a=5)\) \(\Rightarrow 2(5^2+d^2)=58\) \((\because(a-b)^2+(a+b)^2=2(a^2+b^2))\) \(\Rightarrow5^2+d^2=\frac{58}2=29\) \(\Rightarrow d^2=29-5^2=29-25\) \(=4=2^2\) \(\Rightarrow d=2\) \(\therefore a-d=5-2=3\) & \(a+d=5+2=7\) Hence, the required numbers are 3, 5 and 7. Correct option is B) 3, 5, 7

If the sum of three numbers which are in A.P is 15 and the sum of the squares of the extremes is 58, then those numbers are A) 3, 6, 9 B) 3, 5, 7C) 2, 5, 8 D) 2, 6, 7

Answer»

Correct option is (B) 3, 5, 7

Let a - d, a, a+d are required three numbers in A.P.

Given that their sum is 15.

\(\therefore\) (a - d) + a + (a+d) = 15

\(\Rightarrow3a=15\)

\(\Rightarrow a=\frac{15}3=5\)

Also given that the sum of the squares of the extremes is 58.

\(\therefore(a-d)^2+(a+d)^2=58\)

\(\Rightarrow(5-d)^2+(5+d)^2=58\) \((\because a=5)\)

\(\Rightarrow 2(5^2+d^2)=58\) \((\because(a-b)^2+(a+b)^2=2(a^2+b^2))\)

\(\Rightarrow5^2+d^2=\frac{58}2=29\)

\(\Rightarrow d^2=29-5^2=29-25\)

\(=4=2^2\)

\(\Rightarrow d=2\)

\(\therefore a-d=5-2=3\)

& \(a+d=5+2=7\)

Hence, the required numbers are 3, 5 and 7.

Correct option is B) 3, 5, 7