Question

36 is divided into three parts as that they are in A.P sum of squares of number is 464.find the number.

Answer 1

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Answer 2

The three numbers are 8, 12, and 16.

Given,

36 is divided into 3 parts.

The 3 parts are in A. P.

The sum of squares of these numbers (parts) is 464.

To find,

The numbers.

Solution,

It can be seen here that 36 is to be divided into 3 parts such that the parts are in arithmetic progression or A. P.

Firstly, as the parts are in A. P. let these 3 parts be

$(a-d),a,(a+d)$

The sum of these numbers should be 36. So,

$a-d+a+a+d=36$

$\implies 3a = 36$

⇒ a = 12.     ...(1)

Further, it is given that the sum of their squares is 464. That is,

$(a-d)^2+a^2+(a+d)^2 = 464$

Simplifying,

$a^2-2ad+d^2+a^2+a^2+2ad+d^2=464$

$\implies a^2+d^2+a^2+a^2+d^2=464$

$\implies 3a^2 +2d^2=464$

We have, a = 12 from (1), thus,

$3(12)^2 +2d^2=464$

$\implies 3(144) +2d^2=464$

$\implies 2d^2=464-432=32$

$\implies d^2=16$

⇒ d = ±4.

Now, the three numbers can be determined as follows.

For d = 4,

a - d = 12 - 4 = 8,

a = 12,

a + d = 12 + 4 = 16.

⇒ the numbers = 8, 12, 16.

For d = -4,

a - d = 12 - (-4) = 12 + 4 = 16,

a = 12,

a + d = 12 + (-4) = 12 - 4 = 8.

⇒ the numbers = 16, 12, 8.

So, the 3 numbers are 8, 12, and 16 in both the cases whether d is taken +4 or -4.

Therefore, the three numbers are 8, 12, and 16.

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