The number of terms in the sequence 5, 20, 80, 320, ..........., 81920 is:

Concept

In a geometric progression (GP), the nth term is given by an = a·r(n−1), where a is the first term and r is the common ratio. To find how many terms a GP has when its last term L is known, set a·r(n−1) = L and solve for n.

Application

1. Identify the first term and common ratio: a = 5 and r = 20 ÷ 5 = 4 (also 80 ÷ 20 = 4, confirming the ratio).

2. Set the nth term equal to the last term: 5·4(n−1) = 81920.

3. Isolate the power: 4(n−1) = 81920 ÷ 5 = 16384.

4. Write 16384 as a power of 4: 16384 = 47 (since 4·4·4·4·4·4·4 = 16384).

5. Equate exponents: n − 1 = 7, so n = 8.

Cross-check

List the terms directly: 5, 20, 80, 320, 1280, 5120, 20480, 81920. Counting them gives exactly 8 terms, and the 8th term is 81920, which matches the stated last term.

Therefore the sequence has 8 terms.