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Given the following concerning an arithmetic series and a geometric series:

  • The second term of the arithmetic series is the same as the third term of the geometric series. Additionally, the fifth term of the geometric series is the same as the fourteenth term of the arithmetic series.

  • The first term of the arithmetic series is equal to the second term of the geometric series and three times the first term of the said geometric series.

  • The sum of the first four terms of the arithmetic series, $SAP_4$ and the sum of the first three terms of the geometric series, $SGP_3$ are related by the formula

$$SAP_4 \;–\; 4\cdot SGP_3 \;+\; 2 \;=\; 0$$

What is the total of the sum of the first nine terms of the arithmetic series and the sum of the first five terms of the geometric series?

Blue
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Eliza Q
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    Welcome to Math.SE! This is an interesting problem, but you should give us some of your thoughts about solving it, and/or where you got stuck. This information helps answerers target their responses to your experience level and the problem's intended level of difficulty, without wasting anyone's time telling you things you already know. (Plus, it helps convince people that you aren't simply trying to get them to do your homework for you.) Be sure to add such thoughts to the question itself; not everyone reads the comments right away. – Blue Nov 10 '18 at 20:30
  • That's the problem. I dont have any idea how to approach the question even though I'v been looking at some examples. Any hints? – Eliza Q Nov 10 '18 at 21:14
  • Do you know how to write the general form for a term in an arithmetic or geometric series? – Blue Nov 10 '18 at 21:15
  • Are they an = a1 + (n-1) d and an = a12n - 1? – Eliza Q Nov 10 '18 at 21:35
  • Sorry, I dont know how to properly edit the math equations – Eliza Q Nov 10 '18 at 21:35
  • I think I know what you mean. :) I'll write $g$ for the geometric one: $$a_n=a_1+(n-1)d\qquad g_n=g_1 r^{n-1}$$ (Type those as $a_n=a_1+(n-1)d$ and $g_n=g_0 r^{n-1}$. The $s are important.) In any case, you want to translate the various conditions into these forms. As a first pass, write them using $a_n$ and $g_n$; for instance, "second term of arithmetic = third term of geometric" gives $a_2 = g_3$. Do that for all the conditions. Then, go back through and write each $a_n$ in terms of $a_1$ and $d$, and each $g_n$ in terms of $g_1$ and $r$. Then try to find what those values must be. – Blue Nov 10 '18 at 21:45
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    OK, thanks for your input – Eliza Q Nov 10 '18 at 21:47

1 Answers1

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Hint: Have you tried writing out the information given? Start by giving names: for example, $a$ and $d$ are the first term and difference of the arithmetic progression, while $g$ and $r$ are the first term and ratio of the geometric progression.

In particular, you should see that the second item gives you some information pretty quickly.

Thus the arithmetic progression is $\{a, a+d, a+2d, \ldots\}$, and the geometric progression is $\{g, gr, gr^2, \ldots\}$.

So, for example, "the second term of the arithmetic progression is the same as the third term of the geometric series" becomes $a+d = g r^2$, and "the first term of the arithmetic progression is the same as the second term of the geometric series" becomes $a = g r$.

rogerl
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