How many terms are there in the GP 5, 20, 80, 320……20480?

Concept: For a geometric progression (GP) with first term a and common ratio r, the nth term is given by the formula an = a·rn−1. To find how many terms a given GP has, set the last term equal to this nth-term expression and solve for n.

Application:

1. Identify the first term and the common ratio from the series: a = 5, and r = 20/5 = 80/20 = 320/80 = 4.

2. Set the last term equal to the nth-term formula: a·rn−1 = 20480, i.e. 5·4n−1 = 20480.

3. Divide both sides by 5: 4n−1 = 4096.

4. Express 4096 as a power of 4: 46 = 4096, so n − 1 = 6.

5. Solve for n: n = 6 + 1 = 7.

Cross-check: Substituting n = 7 back gives the 7th term as 5 × 46 = 5 × 4096 = 20480, which matches the given last term — confirming the GP has 7 terms.