Question

If x = `sum_(n = 0)^∞a^n`, y = `sum_(n = 0)^∞b^n`, z = `sum_(n = 0)^∞c^n` where a, b , c are in A.P. and |a| < 1, |b| < 1, |c| < 1 then x, y, z are in ______.

Options

• G.P.

• A.P.

• Arithmetic - Geometric Progression

• H.P.

Answer 1

If x = `sum_(n = 0)^∞a^n`, y = `sum_(n = 0)^∞b^n`, z = `sum_(n = 0)^∞c^n` where a, b , c are in A.P. and |a| < 1, |b| < 1, |c| < 1 then x, y, z are in H.P..

Explanation:

x = `sum_(n = 0)^∞a^n = 1/(1 - a)`

⇒ a = `1 - 1/x`

y = `sum_(n = 0)^∞b^n = 1/(1 - b)`

⇒ b = `1 - 1/y`

z = `sum_(n = 0)^∞c^n = 1/(1 - c)`

⇒ c = `1 - 1/z`

a, b, c are in A.P.

⇒ 2b = a + c

`2(1 - 1/y) = 1 - 1/x + 1 - 1/z`

`2/y = 1/x + 1/z`

⇒ x, y, z are in H.P.