really having touble with all that need urgent help!!!
regards,
1)
a geometric series is one of the form: $\displaystyle \sum_{n=1}^{\infty} ar^{n-1}$
where a is the first term, r is the common ratio and n is the current number of the term.
a)
we are given that the second term is 80, $\displaystyle \Rightarrow ar = 80$
we are also told that the fifth term is 5.12, $\displaystyle \Rightarrow ar^4 = 5.12$
$\displaystyle \Rightarrow \frac {ar^4}{ar} = \frac {5.12}{80}$
$\displaystyle \Rightarrow r^3 = 0.064$
$\displaystyle \Rightarrow r = 0.4 \mbox{ or } \frac {4}{10}$
since $\displaystyle ar = 80 \mbox{ we have } a = \frac {80}{r} = \frac {80}{ \frac {4}{10}} = 200$
so the first term is 200, and the common ration is $\displaystyle \frac {4}{10}$
b)
For a geometric series with $\displaystyle \left| r \right| < 1$, the sum to infinity is given by: $\displaystyle S_{ \infty} = \frac {a}{1 - r}$
So here, $\displaystyle S_{ \infty} = \frac {200}{1 - \frac {4}{10}} = 333 \frac {1}{3}$
c)
The sum of the first $\displaystyle n$ terms of a geometric series is given by $\displaystyle S_{n} = \frac {a(1 - r^n)}{1 - r}$
So $\displaystyle S_{ \infty} - S_{14} = 333 \frac {1}{3} - \frac {200(1 - 0.4^{14})}{1 - 0.4} = 0.0008947 = 9 \times 10^{-4}$