# G.P.

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• Feb 15th 2010, 02:49 AM
Sunyata
G.P.
Show that there are 3 geometrical progression in which the second term is - 4/3 and the sum of the first three terms is 28/9. SHow that one of these series is convergent and, in this case, find the limit of its sum.

How on EARTH do i do this question???

x - 4/3 + y = 28/9...
• Feb 15th 2010, 03:07 AM
Prove It
Quote:

Originally Posted by Sunyata
Show that there are 3 geometrical progression in which the second term is - 4/3 and the sum of the first three terms is 28/9. SHow that one of these series is convergent and, in this case, find the limit of its sum.

How on EARTH do i do this question???

x - 4/3 + y = 28/9...

Since this is a geometric progression, you know that $\displaystyle t_{n + 1} = rt_n$.

Therefore $\displaystyle r = \frac{t_{n + 1}}{t_n}$.

So here $\displaystyle r = \frac{-\frac{4}{3}}{x}$ and $\displaystyle r = \frac{y}{-\frac{4}{3}}$.

Therefore $\displaystyle \frac{-\frac{4}{3}}{x} = \frac{y}{-\frac{4}{3}}$

$\displaystyle y = \frac{-\frac{4}{3}\left(-\frac{4}{3}\right)}{x}$

$\displaystyle y = \frac{\frac{16}{9}}{x}$

$\displaystyle y = \frac{16}{9x}$.

Therefore, the sum of the first three terms is

$\displaystyle x - \frac{4}{3} + \frac{16}{9x} = \frac{28}{9}$.

Solve for $\displaystyle x$ and then you can find $\displaystyle y$.
• Feb 15th 2010, 03:10 AM
Sunyata
What does it mean by convergent?
• Feb 15th 2010, 03:16 AM
Prove It
It means that if you had an infinite number of terms, if you were to add them up, you would find that the sum tends to a number, rather than tending to $\displaystyle \infty$.