Find the sum of this geometric series?

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1 Answer
Dean R.
Apr 24, 2018

# sum_{ n=1}^6 5 (2)^{n+1} = 1260 #

Explanation:

Sometimes brute force is the easiest way.

# sum_{ n=1}^6 5 (2)^{n+1} #

# = 5 sum_{ n=1}^6 (2)^{n+1} #

# = 5( 4 + 8 + 16 + 32 + 64 + 128 ) #

# = 5(252) #

#= 1260 #

I'm just kidding, They want you to apply the formula

#sum_{k=0}^{n-1} a r^k = {a(1- r^n)}/{1-r}#

We don't really have to fuss getting the indices to match this form. We just say the first term is #a=20,# the ratio #r=2,# and the count #n=6,# so the sum is

# sum_{ n=1}^6 5 (2)^{n+1} = {20 (1 - 2^6)}/{1-2} = 20(2^6-1)=20(63)=1260#

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