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

1 Answer
Shwetank Mauria
Mar 28, 2016

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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