Answer:
n = 7
number of terms is 7
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