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Coming from a finance guy, I understand how AP and GP work. However, I came upon a problem that combines the two and was stuck. Here it goes.

Given first term of AP and GP$=4$, and common ratio of GP is $8$ less than common difference of AP. The ratio of $3$rd term of AP to $3$rd term of GP is $7:16$. What is the common difference and common ratio?

I tried it this far:

given common ratio$=r$

difference$=d$

so $r=d-8$

for the AP; 3rd term$=4+2d$

and GP$=4(d-8)(d-8)$

so $(4+2d) : 4(d-8)(d-8) =7:16$

and I got this far..

some headway please..

N. F. Taussig
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    Welcome to MSE. It will be more likely that you will get an answer if you show us that you made an effort. This should be added to the question rather than in the comments. – José Carlos Santos Sep 30 '18 at 10:30
  • You correctly found that $$4 + 2d : 4(d - 8)(d - 8) = 7 : 16$$ which means $$\frac{4 + 2d}{4(d - 8)(d - 8)} = \frac{7}{16}$$ which reduces to a quadratic equation in $d$. Can you proceed from here? – N. F. Taussig Oct 07 '18 at 17:20

1 Answers1

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Here, assume that the AP is :

$$a, a+d, a+2d, ...$$

and, the GP is :

$$ a, ar, ar^2 ,...$$

Then from the given data -

$a = 4$, $ r = d - 8$ ( or $d = r + 8$) and $\frac {a + 2d}{2r^2} = \frac{7}{16}$.

The third equation above simplifies as,

$\frac{4+2d}{4r^2} = \frac{2+d}{2r^2} = {7\over 16}$

$\implies \frac{r+10}{2r^2} = frac{7}{16}$ ( by the second equation)

$\implies 7r^2 - 4r - 40 = 0$

or, $r = \frac{4 \pm \sqrt{16 + 4.7.40}}{2.7} = \frac{4\pm \sqrt{1136}}{14}$ (using the quadratic formula)

Thus, $r = 2.693$ or $-2.121$ (approximately)

Using the second equation above, $d = 10.693$. or $5.8782$ (approx.)

user0
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