Question

The sum of the three natural numbers in an AP is 75 and the sum of their squares is 1893. What is the smallest of the three numbers​

Answer 1

Given:

The sum of three natural number = 75

there square =1893

To find:

number=?

Solution:

let given number= (a-d), a, (a+d)

$\Rightarrow \bold{(a-d)+a+(a+d)= 75}\\\\\Rightarrow (a-d+a+a+d)= 75\\\\\Rightarrow (a+a+a)= 75\\\\\Rightarrow 3a= 75\\\\\Rightarrow a= \frac{75}{3}\\\\\Rightarrow a= 25$

$\Rightarrow \bold{(a-d)^2+a^2+(a+d)^2=1893}\\\\\Rightarrow a^2+d^2-2ad+a^2+a^2+d^2+2ad=1893\\\\\Rightarrow a^2+d^2+a^2+a^2+d^2=1893\\\\\Rightarrow 3a^2+2d^2=1893\\\\\Rightarrow 3(25)^2+2d^2=1893\\\\\Rightarrow 3\times 625+2d^2=1893\\\\\Rightarrow 1,875+2d^2=1893\\\\\Rightarrow 2d^2=1893-1,875\\\\\Rightarrow 2d^2= 18\\\\\Rightarrow d^2= 9\\\\\Rightarrow d^2= 3^2\\\\\Rightarrow d= 3$

The calculated number:

$\Rightarrow (a-d), a,(a+d)\\\\\Rightarrow (25-3), 25,(25+3)\\\\\Rightarrow 22, 25, 28$

The final value is 22, 25, 28

Answer 2

Answer:

22

Step-by-step explanation:

Let the required numbers are a - d, a, a + d.

   Their sum = 75       [given]

⇒ (a - d) + a + (a + d) = 75

⇒ 3a = 75

⇒ a = 25

Given,    sum of their squares is 1893

⇒ (a - d)² + a² + (a + d)² = 1893

⇒ (a - d)² + (a + d)² + a² = 1893

⇒ 2(a² + d²) + a² = 1893

⇒ 3a² + 2d² = 1893

⇒ 2d² = 1893 - 3(25)²            [a = 25]

⇒ d² = 9

⇒ d = 3   or   -3

   Therefore,  the given AP is

a - d = 25 - 3       or  25 - (-3)

        = 22            or    28

a = 25

a + d = 25 + 3     or  25 - 3

        = 28         or 22

Hence the AP is 22, 25, 28  or  28, 25, 22.   In both the cases, smallest term number is 22