Need help with geometric series: why is it k+1?

[genyme]

Hi,

I'm having a bit of trouble with understanding geometric series and was wondering if someone can help me out?

E.g 1 + 2 + 4 + 8 + ... 2^k

The answer is 2^(k + 1) - 1, can someone help explain why that is?

I can see the pattern: 2^0, 2^1, 2^2, ...
So to get the n = 3 term, then 2^3-1 = 2^2 = 4 --> which would be 2^k-1. But why is is k+1? Also where did the minus 1 come from at the end?

Thanks!

 

[mmm4444bot]

I'm having a bit of trouble with understanding geometric series …

1 + 2 + 4 + 8 + ... 2^k

The answer is 2^(k + 1) - 1, can someone help explain why that is?

So to get the n = 3 term, then 2^(3-1) = 2^2 = 4 --> which would be 2^(k-1). But why [does the answer show] k+1? Also where did the minus 1 come from at the end?

The kth term is 2^k, not 2^(k-1). They are starting the index at 0:

a₀ = 1
a₁ = 2
a₂ = 4
a₃ = 8
…
a_k = 2^k

Are you confusing the formula for the sequence with the formula for the sum (i.e., the series)?

a₀ = first term
a_n = a₀ * rⁿ

_{(n = 0 to k)} ∑ a_n = a₀ * (1 - r ^k+1 ) / (1 - r)

 

[genyme]

The kth term is 2^k, not 2^(k-1). They are starting the index at 0:

a₀ = 1
a₁ = 2
a₂ = 4
a₃ = 8
…
a_k = 2^k

Are you confusing the formula for the sequence with the formula for the sum (i.e., the series)?

a₀ = first term
a_n = a₀ * rⁿ

_{(n = 0 to k)} ∑ a_n = a₀ * (1 - r ^k+1 ) / (1 - r)

It should be the sum (series). Why did you use r^(k+1) - shouldn't it be r^k?

 

[mmm4444bot]

Why did you use r^(k+1) -- shouldn't it be r^k?

It's r^k, when the index starts at 1.

It's r^(k+1), when the index starts at 0.

In the given sequence, the index starts at 0.

If, instead, the index had started at 1, they would have given the kth term as 2^(k-1), instead of 2^k. :cool: