1. The smallest number is

Question

Four different integers form an increasing AP. One of these numbers is equal to the sum of the squares of the other three numbers. Then

1.  The smallest number is

(A)  -2

(B)  0 

(C)  -1

 (D)  2

2.  The common difference of the four numbers is

(A)  2

(B)  1

(C)  3

(D)  4

3.  The sum of all the four numbers is

(A)  10

(B)  8

(C)  2

(D)  6

Answer 1

Correct answer  1.(C)  2. (B) 3. (C)

1.  Let the four integers be a - d, a, a + d and a + 2d where, a and d are integers and d > 0. Since

a + 2d = (a - d)² + a² + (a + d)²

⇒  2d² - 2d + 3a² - a = 0   .....(1)

Therefore,

Since, d is positive integer 

Therefore,

Since, a is an integer, therefore a = 0. Put in Eq. (2) we get d = 1 or 0.

But, since d > 0, therefore, d = 1.

The smallest number is - 1

Therefore, the four numbers are: -1, 0, 1, 2 

2. The common difference of the four numbers is d = 1.

3. The sum of all the four numbers is = -1+ 0+ 1+ 2 = 2.