Question

The sum of the three numbers in A.P. is 36. When the numbers are increased by 1, 4, 43 respectively, the resulting numbers are in G.P. Find the numbers.

Answer 1

Answer:

Step-by-step explanation:

Answer 2

Given:

The sum of the three numbers in A.P. is 36

When the numbers are increased by 1, 4, 43 respectively, the resulting numbers are in G.P

To Find:

The three numbers

Solution:

Let the three numbers be a, b, and c

Since they are in A.P.

⇒ 2b = a + c     - (1)

Also given that a + b + c = 36     - (2)

Substituting (1) in equation (2),

2b + b = 36

or 3b = 36

or b = 12

From equation (1), ⇒ a + c = 24

or a = 24 - c        (3)

Also, since (a+1), (b+4), and (c+43) are in G.P.

⇒ (b+4)² =  (a+1) X (c+43)

We know that, b + 4 = 12 + 4 = 16

So, 16 X 16 = (a+1) X (c+43)

Substituting the value of a from equation (3) we get

256 = (24-c+1) X (c+43)

or (25 - c ) X (c + 43) = 256

Opening the brackets and solving,

25c + 1075 -c² - 43c = 256

or 1075 -c² -18c = 256

or c² + 18c - 819 = 0

Solving the quadratic equation we get,

c = 21          or             c = -39

From equation (3),

If c = 21 ⇒ a = 3

If c = -39 ⇒ a = 63

Hence, the three numbers are 3, 12, and 21 or 63, 12, and -39.