Honors Algebra 2 Sequences Question

Question

The sum of the first 10 terms of a geometric sequence is 4-2^(-8). If the common ratio is 1/2, find the first term of the sequence. Use this to determine the 11th term of the sequence.

Answer 1

The formula for a geometric sum is a(1 - rⁿ)/(1-r), where a is the first term, r is the ratio, and n is the number of terms.

Substituting r = 1/2 and n = 10we get

Thus, a(1 - (1/2)¹⁰)/(1 - 1/2) = a((1 - (1/2)¹⁰)/(1/2) = 2a(1 - (1/2)¹⁰) = 2a(1 - 2^-10)

As the sum is 4 - 2^-8 = 4(1 - 1/4 * 2^-8) = 4(1 - 2^-10)

4(1 - 2^-10) = 2a(1 - 2^-10),

4 = 2a

a = 2

The first term is 2.

The n'th term is ar^n-1 = 2 * (1/2)^11-1 = 2 * (1/2)¹⁰ = 2^-9