The number of terms in the sequence 5, 20, 80, 320,...................., 5120…

Concept

In a geometric progression (GP) each term is obtained by multiplying the previous term by a fixed common ratio r. If the first term is a, the nth term is given by the formula aₙ = a·r raised to the power (n−1). To find how many terms a finite GP has, set its general term equal to the known last term and solve for n.

Application

1. Identify a and r. The first term is a = 5, and dividing consecutive terms (20÷5, 80÷20, 320÷80) gives the common ratio r = 4.

2. Write the general term: an = 5·4(n−1).

3. Set it equal to the last term 5120: 5·4(n−1) = 5120.

4. Divide both sides by 5: 4(n−1) = 1024.

5. Express 1024 as a power of 4: 1024 = 45, so 4(n−1) = 45.

6. Equate the exponents: n − 1 = 5, hence n = 6.

Cross-check

List the terms directly: 5, 20, 80, 320, 1280, 5120. Counting them confirms there are exactly 6 terms, and the sixth term equals the given last value 5120. So the sequence has 6 terms.