I study maths as a hobby.
I have come across this question:
The sum of the first n terms of a geometric series is $8 – 2^{3 – 2n}$. Find the first term of the series, its common ratio and its sum to infinity.
Now I know that the formula for a geometric series is
$S_n = \frac{a(1 – r^n)}{1 – r}$ and that
$8 – 2^{3 – 2n} = 2^3 – \frac{2^3}{2^{2n}}$ but I cannot see how that leads me to the first term.
Best Answer
You need to put in values, starting from 1,2,3 and so on. Since it asks for first term, use $n=1$.
Also for finding the sum to infinity, you need to use another formula:
$S$$\infty$ = $a/(1-r)$
where $a$ is the first term of the geometric progression and $r$ is the common ratio.
Or, like @Rob Arthan mentioned, you can also calculate the sum to infinity by using :
$\lim_{n\to\infty} 8 - 2^{3-2n}$
This would be more useful as you wouldn't have to calculate the common ratio at all.