# Find the largest natural number, k, where the difference between the terms of these two series is larger than 14.The kth term of a series, Sk= a*1-r^k/1-r, is the sum of the first k terms o fthe...

Find the largest natural number, k, where the difference between the terms of these two series is larger than 14.

The kth term of a series, Sk= a*1-r^k/1-r, is the sum of the first k terms o fthe underlying sequence. The difference between the nth terms of two particular series is greater than 14 for some values of nEN. The series with general term tn= 100(11/17)^n-1 begins largeer 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.

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(1) The `n^(th)` term of the underlying sequence for the first series is given as `t_n=100(11/17)^(n-1)` . The approximate values for the terms in the sequence are: 100,64.706,41.865,27.091,...

The series which has this underlying sequence has values given by `S_k=100(1-(11/17)^k)/(1-(11/17))` , and the first few values for this series are 100,164.71,206.57,233.67,251.2,...

(2) The `n^(th)` term of the underlying sequence for the second series is given as `t_n=50(14/17)^(n-1)` . The approximate values for the first few terms of this sequence are: 50,41.176,33.91,27.926,22.998,18.939,...

The series which has this underlying sequence has values given by `S_k=50(1-(14/17)^k)/(1-14/17)` , and the first few values for this series are

50,91.176,125.09,153.01,176.01,194.95,...

(3) We are asked to find the largest `k in NN` such that the difference between the corresponding entries of the two series is greater than 14. Using a computer algebra system (spreadsheet, graphing calculator, etc...) we find that ` `for any `k<=15` the difference is greater than 14.

**Thus k=15 is the required answer.**

(Getting an exact answer requires solving `(11/17)^k+(14/17)^k>21/425` , which is difficult)

`100(1-(11/17)^k)/(1-11/17)-50(1-(14/17)^k)/(1-14/17)>14`

`100(1-(11/17)^k)/(6/17)-50(1-(14/17)^k)/(3/17)>14`

`850/3-850/3(11/17)^k-[850/3-850/3(14/17)^k]>14`

`(11/17)^k+(14/17)^k>42/850=21/425`

**Sources:**