Question

The sum or difference of two G.P.s, is again a G.P.

Options

• True

• False

Answer 1

This statement is False.

Explanation:

Let us consider two G.P.’s

a₁, a₁r₁, a₁r₁², a₁r₁³ ... a₁r₁^n-1

And a₂, a₂r₂, a₂r₂², a₂r₂³, ... a₂r₂^n-1

Now Sum of two G.Ps

`(a_1 + a_2) + (a_1r_1 + a_2r_2) + (a_1r_1^2 + a_2r_2^2)  ...`

Now `T_2/T_1 = (a_1r_1 + a_2r_2)/(a_1 + a_2)`

And `T_3/T_2 = (a_1r_1^2 + a_2r_2^2)/(a_1r_1 + a_2r_2)`

But `(a_1r_1 + a_2r_2)/(a_1 + a_2) ≠ (a_1r_1^2 + a_2r_2^2)/(a_1r_1 + a_2r_2)`

Now let us consider the difference G.P’s

`(a_1 - a_2) + (a_1r_1 - a_2r_2) + (a_1r_1^2 - a_2r_2^2)`

∴ `T_2/T_1 = (a_1r_1 - a_2r_2)/(a_1 - a_2)`

And `T_3/T_2 = (a_1r_1^2 - a_2r_2^2)/(a_1r_1 - a_2r_2)`

But `T_2/T_1 ≠ T_3/T_2`