Sequences and series - try again

[oilersforever72]

Sequences and series - try again :)

Hi, I'm going to try to post this question again, hopefully it is more clear this time. I'm not sure how to approach this question, or really, what this question is asking me!

Homework Statement

The k-th term of a series, Sk = a*[(1-(r^k))/(1-r)], is the sum of the first k terms of the underlying sequence.

(Note: This is a general formula that I remember from grade 12 math where a is the first term in a sequence, and r is the constant ratio between subsequent terms. Correct me if I'm wrong .)

The difference between the n-th terms of two particular series is greater than 14 for some values of n (where n is a Natural number). The series with general term tn = 100[(11/17)^(n-1)] begins larger than the second series with general term tn = 50[(14/17)^(n-1)]. Find the largest natural number, k, where the difference between the terms of these two series is larger than 14.

Homework Equations

The Attempt at a Solution

I'm hooped.

 

[arildno]

Try to set up an inequality you hope can be solved for k!

 

[oilersforever72]

Hmm... Okay, what is the problem actually asking for?

FIRST Sk: So given the first general term, I can plug it into the Sk formula to get Sk = 100[1-(11/17)^k)]/[(1-(11/17)].

SECOND Sk: Given the second general term, I can plug it into the Sk formula again to get Sk = 50[1-(14/17)^k]/[1-(14/17)].

Is it asking me to solve for k by subtracting the second Sk from the first Sk and setting it equal or greater to 14?

 

[oilersforever72]

Hmm...

 

[arildno]

Indeed it is!
However, I know of any simple formula to compute this, with a subtraction between power function with different bases&multiplicative factors yielding a non-zero answer.

Therefore, I think you just have to plug different k-values into your expression until you get the right answer.

Dumb exercise.

 

[oilersforever72]

That's exactly why this question confused me. My work didn't turn out pretty, then I psyched myself out by thinking I must have read the question wrong. Thanks man.

 

[HallsofIvy]

One of the things that is confusing here, just as it was when you posted this same question before, is your use of the phrase "kth term of the series" when you started by giving a formula for the kth partial sum (of a geometric series). As I told you before, the "kth term of the series" $\Sigma a_n$ is a_k. The sum a₁+ a₂+ ... + a_k is the "kth partial sum".

You are given two geometric series, $\Sigma a_n$ and $\Sigma b_n$ . Do you want to find k so that the difference of the kth terms, a_k- b_k, is greater than 14, or so that the difference of the kth partial sums, $\Sigma_{i=1}^k (a_n- b_n)$ is greater than 14?

 

[oilersforever72]

Grrr... Argh... It's a retarded question.