Question

if three numbers a,8,b,(a≠b) are in gp and a,b,-8 are in A.p .then values of a and b resp. are​

Answer 1

Answer:

The value of a is 16, the value of b is 4 .

Step-by-step explanation:

Given:

• Three numbers a, 8, b are in G.P
• The numbers a, b, -8 are in A.P

To Find:

• Values of a and b

Solution:

If a , 8 , b are in G.P we know that.

b/8 = 8/a

Hence,

ab = 8 × 8

ab = 64

b = 64/a-----(1)

Now if they are in A.P we know that,

- 8 - b = b -a

-8 + a = 2b

Substitute the value of a from equation 1

-8 + a = 2 (64/a)

-8 + a = 128/a

-8a + a² = 128

a² - 8a - 128 = 0

Factorising by splitting the middle term,

a²- 16a -+ 8a - 128 = 0

a (a - 16) + 8 (a - 16) = 0

(a - 16) (a + 8) = 0

Either

a + 8 = 0

a = -8

Or

a - 16 = 0

a = 16

Case 1:

If a = -8

b = 64/-8

b = -8

Hence if a is -8, the value of b is -8.

This is not possible because a ≠ b is given.

Case 2 :

If a = 16

b = 64/16

b = 4

Hence if a is 16, the value of b is 4.