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

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

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Answer 1 · 2016-03-28T08:32:10

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

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

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