Sum of a Finite Geometric Series is not Consistent

Question

Hello,

I'm very confused by a sum of a geometric series as solving it tow different ways gives me different results but I can't tell which way is wrong.

3

∑ 7*(1/10)^2n

n=1

a1 = 7*(1/10)^2=0.07

a2 = 7*(1/10)^4=0.0007

a2 = 7*(1/10)^6=0.000007

sum = 0.070707

But the sum of a finite geometric series is given as:

sum = (a1*(1-r^n))/(1-r) = (0.07*(1-(1/10)^3))/(1-1/10) = 0.0777

when r =/= 1

How can both of these be right? Any help is appreciated.

Answer 1

Your common ratio is the first is 1/100 not 1/10

The common ratio in the second is 1/10

Answer 2

sum = (a1*(1-r^n))/(1-r) = (0.07*(1-(1/10)^3))/(1-1/10) = 0.0777

r = 0.01 = 1/100

Answer 3

In your case, the geometric equation is 7r²(1-r⁶)/(1-r²)

This gives the first answer of .070707

This is seen by doing the following:

Let S = r² + r⁴ + r⁶ then multiply S by r² to get

r²S = r4 + r⁶ + r⁸ now subtract the two to get

------------------------

S - r²S = r² - r⁸ = S(1-r²) Therefore, S = (r² - r⁸)/(1-r²) = r²(1-r⁶)/(1-r²)

substituting .1 for r, we get S = .010101 multiply by 7 to get .070707

(The 7 is a constant & comes out of the summation sign)!