Question

If a, b, c are in G.P and a+x,b+x,c+x are in H.P, then what is the value of x(a,b,c are distinct)​

Answer 1

solution attached

x = b

Answer 2

Answer:

B

Step-by-step explanation:

a, b, c be in gp

Then b²=ac

a+x, b+x, c+x in hp

Then 1/a+x, 1/b+x, 1/c+x are in ap

1/b+x - 1/a+x = 1/c+x - 1/b+x

a-b/ab+ax+bx+x² = b-c/bc+bx+cx+x²

(a-b) (bc+bx+cx+x²) =(b-c) (ab+ax+bx+x²)

abc+abx+acx+ax²-b²c-b²x-bcx-bx² = ab²+abx+b²x+bx²-abc-acx-bcx-cx²

2abc+acx+ax²-b²c-b²x-bx²-ab²-b²x-bx²+acx+cx = 0

2abc+2acx-2bx²-2b²x-ab²-cb²+ax²+cx² = 0

2bx²-ax²-cx²-2acx+2b²x-2abc+ab²+cb² = 0

(2b-a-c)x² +(2b²-2ac) x - 2abc+ab²+cb² =0

b²=ac

(2b-a-c) x²+(2ac-2ac)x + b²(a+c-2b) = 0

(2b-a-c) x² - b²(2b-a-c) = 0

( 2b-a-c) ( x²-b²) = 0

( 2b-a-c) (x+b) (x-b) = 0

The value of x is +b, -b

So we can say that the value of x = b