in terms of a, what is the value of sigma...

Question

 7                6

 ∑     ar^k - ∑   ar^k  if r=2?  show work.

k=1            k=0

 

Answer 1

Hi Peter,

 

There are two geometric series with ratios equal to 2.

We can place the constant a in front of the sigmas in both series.

Both have 7 terms but one starts at 0 while the other starts at 1.

 

The sum of the first series is

S1 = a₁*(1-rⁿ)/(1-r)

a1 = 2 ,  n=7

S1 = 2*127

 

The sum of the second series is

S2 = a1*(1-rⁿ)/(1-r)
a1 =1 , n=7
S2 = 127

 

S1 - S2 = a(2*127 - 127)

= 127a