Question

Which term of the GP 2,1,1/2,1/4,.... Is 1/1024?

Answer 1

a×r^n-1=1/1024
2×(1/2)^n-1=1/1024
(1/2)^n-1=1/2048
(1/2)^n-1=(1/2)^11
so n-1=11
so n=12

Answer 2

Geometric Progression: If some numbers are arranged in such a way that the ratio of a number and its previous number or the ratio of a number and its next number is always same, then the numbers are said to be in GP, and the ratio is said to be common ratio.

To find the General Term of the GP:

If a be the first term and r be the common ratio, then the GP be

a, ar, ar², ar³, ...

∴ the n-th term, tₙ = a rⁿ⁻¹

Solution:

The given GP is

2, 1, 1/2, 1/4, ...

The first term (a) = 2

& the common ratio (r) = 1/2

Let 1/1024 be the n-th term of the GP.

Then a * rⁿ⁻¹ = 1/1024

or, 2 * (1/2)ⁿ⁻¹ = 1/1024

or, 2 * (1/2)ⁿ * (1/2)⁻¹ = 1/1024

or, 2 * (1/2)ⁿ * 2 = 1/1024

or, (1/2)ⁿ = 1/4096

or, (1/2)ⁿ = (1/2)¹²

Equating the exponents from both sides, we get

n = 12

∴ 1/1024 is the 12th term of the given GP.