The common ratio of a geometric progression is 1/2 , the fifth term is 1/80 , and the sum of all of its terms is 127/320 . find the number of terms in the progression.

n = 7

number of terms is 7

Step-by-step explanation:

The common ratio is 1/2 of a geometric progression . The fifth term = 1/80

sum of all it terms = 127/320.

The number of term in the progression can be computed as follows:

let us get the first term

nth term = arⁿ⁻¹

where

a = first term

n = number of terms

r  = common ratio

nth term = arⁿ⁻¹

fifth term = 1/80

1/80 = a × 1/2⁵⁻¹

1/80 = a × 1/2⁴

1/80 = a/16

cross multiply

16 = 80a

divide both sides by 80

a = 16/80

a = 1/5

The sum of all terms = 127/320

sum = a(1 - rⁿ)/1 - r

127/320 = 1/5 (1 - 1/2ⁿ)/ 1 - 1/2

127/320 = 1/5(1 - 1/2ⁿ)  / 1/2

127/320 = 2/5(1 - 1/2ⁿ)

127/320 = 2/5 - 2/5 × 1/2ⁿ

collect like terms

127/320 - 2/5 = - 2/5 × 1/2ⁿ

(127 - 128)/320 = - 2/5 × 1/2ⁿ

-1/320 =  - 2/5 × 1/2ⁿ

multiply both sides by -5/2

-1/320 × - 5/2 =  1/2ⁿ

5/640 =  1/2ⁿ

1/128 =  1/2ⁿ

1/2⁷ =  1/2ⁿ

both sides have same base

n = 7

number of terms  = 7

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