The number of terms in the sequence 5, 20, 80, 320, _______, 20480 is:

Concept

A geometric progression (GP) is a sequence in which each term equals the previous term multiplied by a fixed non-zero common ratio r. With first term a, the n-th term is Tn = a·rn−1. The total number of terms is found by setting this formula equal to the known last term and solving for n; the count does not depend on how many terms are written out versus left blank.

Application

1. Find the first term and common ratio: a = 5 and r = 20 ÷ 5 = 4 (confirm: 80 ÷ 20 = 4 and 320 ÷ 80 = 4).

2. The last term of the sequence is 20480, so set Tn = 20480: 5·4n−1 = 20480.

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

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

5. Equate the exponents: n − 1 = 6, hence n = 7. The blank simply hides the intermediate terms; the formula fixes the total at 7.

Cross-check

Listing every term of the GP from 5 up to 20480 confirms the count:

• T1 = 5

• T2 = 20

• T3 = 80

• T4 = 320

• T5 = 1280 (an omitted middle term)

• T6 = 5120 (an omitted middle term)

• T7 = 20480

Counting from T1 to T7, the sequence contains 7 terms.