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. the difference between the "n"th terms of two particular series is greater than 14 for some values of n (is an element of) N. 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 then 14

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To find the largest natural number "k" where the difference between the terms of the two series is larger than 14, we need to compare the terms of the two series and find the value of "k" where the difference exceeds 14.

Let's break down the problem step by step:

First, let's find the general term of both series using the given information:

Series 1:
The general term of series 1 is given by t₁ = 100(11/17)^(n-1)

Series 2:
The general term of series 2 is given by t₂ = 50(14/17)^(n-1)

Now, we want to find the largest "k" where the difference between t₁ and t₂ is larger than 14. Let's set up the inequality:

t₁ - t₂ > 14

Substituting the equations for t₁ and t₂:

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

To solve this inequality, we can simplify it further:

(11/17)^(n-1) - (7/17)^(n-1) > 7/25

Now, we need to find the largest "k" which satisfies this inequality. Let's find a pattern to simplify the left side and try some values of "k":

When "k" = 1:
(11/17)^(1-1) - (7/17)^(1-1) = 1 - 1 = 0

When "k" = 2:
(11/17)^(2-1) - (7/17)^(2-1) = 11/17 - 7/17 = 4/17

When "k" = 3:
(11/17)^(3-1) - (7/17)^(3-1) = (11/17)^2 - (7/17)^2

We can notice a pattern that this difference is decreasing as "k" increases. To find the largest "k" where the difference exceeds 14, we need to continue checking values until we find the one that satisfies the inequality.

Using a calculator or a spreadsheet program, let's calculate a few values:

When "k" = 4:
(11/17)^(4-1) - (7/17)^(4-1) = 0.080

When "k" = 5:
(11/17)^(5-1) - (7/17)^(5-1) = 0.066

When "k" = 6:
(11/17)^(6-1) - (7/17)^(6-1) = 0.054

When "k" = 7:
(11/17)^(7-1) - (7/17)^(7-1) = 0.044

When "k" = 8:
(11/17)^(8-1) - (7/17)^(8-1) = 0.036

From here, we can see that the difference is decreasing and becomes less than 14. Hence, the largest natural number "k" where the difference exceeds 14 is k = 7.

Therefore, the answer is k = 7.