Question

If a, b, c are in G.P. and a, x, b, y, c are in
A.P., prove that:

(i)
1/x + 1/y = 2/b

(ii)
a/x + c/y = 2​

Answer 1

Step-by-step explanation:

If a,x,b,y,c are in AP then a,b,c are also in AP. But a,b,c are also in GP. Therefore a=b=c

(proof below).

x=12(a+b)

and y=12(b+c) and so a=b=c=x=y

and the two required results are trivially true.

Proof that three numbers that are in both AP and GP must all be equal:

Let the numbers be a,ar,ar2

, since they are in GP. Since they are also in AP, the differences must be equal:

ar−a=ar2−ar

ar2–2ar+a=0

a(r−1)2=0

Therefore either a=0

(meaning that all three numbers are zero) or r=1 (meaning that all three numbers are a)■

hope it helpss