Geometric Progression Word Problem

Sep 2015
11
0
Ohio
Q: The sum of the first and last terms of a G.P. of fourteen real terms is 7. The fifth term is the mean proportional between the second and last terms. Find the third term.

All I have been able to come up with thus far are the equations:

t1 + t14 = 7
and
t2/t5 = t5/t14

Then I plugged in a for t1, ar for t2, ar4 for t5, and ar13 for t14

I tried to work through the two equations from here, solving one for a, then plugging that into the other equation to get r, then using r to get a, but I haven't been able to get an answer.

Any help is greatly appreciated.
 
Last edited:

romsek

MHF Helper
Nov 2013
6,642
2,993
California
a geometric progression $\{g_k\}$ is one where $\dfrac{g_{k+1}}{g_k} = r$, a non-zero constant.

so it will look like

$\{a, a r, a r^2, \dots, a r^{k-1}, \dots\, a r^{n-1}\}$

your first equation says

$a + a r^{13}=7$

and your second equation says

$a r^4 = \sqrt{(a)(a r^{13})}$

$ a r^4 = a r^7$

$r=1$

using this in the first equation

$2a=7$

$a = \dfrac 7 2$
 
  • Like
Reactions: 1 person
Sep 2015
11
0
Ohio
Wow, I have no idea how I was oblivious enough to not realize that r=1... I had ar4 = ar7 but somehow didn't pick up on the obvious fact that r could only equal one.

Thanks.