the first 3 terms of a geometric progression are4,2,1. find the 20th term, expressing your answer as a power?

Question

of 2. find also the sum to infinity of this progression.

Answer 1

Recall that a geometric progression utilizes the following pattern.

(a, ar, ar^2, ar^3, ar^4, .... )

For "a" being the first term and r being the ratio.

The general pattern is

ar^(n - 1)

The first term "a" is obviously 4. We want the 20th term, so n = 20. Finally, to get r, we just divide any number in the sequence by its previous term. 2/4 = 1/2, so r = 1/2.

The 20th term is

(4)(1/2)^(20 - 1)
(4) (1/2)^(19)

But 4 = 2^2, so

(2^2) (1/2)^(19)

Note that (1/2) is equal to 2^(-1), so

(2^2) (2^(-1))^(19)
(2^2) (2^(-19))

2^(-17)

If we wanted the sum to infinity, the formula would be

S = a/(1 - r)
S = 4/(1 - [1/2])
S = 4/(1/2) = 8

Answer 2

4,2,1,1/2, ..4*(1/2)^n.....
The 20th term occurs when n=19 so
20th term = 4*(1/2)^19 = 1/2^17 = 2^-17