Question

How do you write an explicit formula for the sequence 40, 20, 10, 5, 2.5...?

Answer 1

Explicit formula for the `n^(th)` term `T_n` of the series is `40(1/2)^(n-1)` and sum of the series `S` is given by `80(1-1/(2^n))`

Explanation:

In the series `{40, 20, 10, 5, 2.5...}`, the ratio of a term to its preceding term is `20/40=10/20=5/10=2.5/5=...=1/2` always constant. Hence it is a geometric series of type `{a,ar,ar^2,ar^3,....}` with first term `a=40` and `r=1/2`.

As is apparent `n^(th)` term of the series

`T_n=ar^(n-1)` or `40(1/2)^(n-1)`.

Let the sum of the series be `S`, then

`S=a+ar+ar^2+ar^3+....+ar^(n-1)` -------(A)

and multiplying both sides by `r`, we get

`rS=ar+ar^2+ar^3+ar^4....+ar^n` -------(B)

Subtracting (B) from (A)

`(1-r)S=ar^n-a=a(1-r^n)`

Hence `S=(a(1-r^n))/(1-r)=40(1-1/(2^n))/(1-1/2)=40(1-1/(2^n))/(1/2)=80(1-1/(2^n))`