Please Help!! Will Medal!!!

The following series is a geometric series.

Question

Please Help!! Will Medal!!!

The following series is a geometric series.

anonymous · Answer

attachment

kainui · Answer

What do you think, and why? Tell me what you think a geometric series is.

anonymous · Answer

geo series=ratio of numbers going in some order... Don't really know.
And I really don't know.

kirbykirby · Answer

Verify if you can write the sum in the form:
$$\large \sum_{i=1}^nar^{i-1} $$

kirbykirby · Answer

$a$ is a constant, and $r$ is the common ratio which is obtained by dividing on term in the sequence by one previous term.

kirbykirby · Answer

hint: can you write $\large 2^{2i}$ in a different way?

anonymous · Answer

4i?

kirbykirby · Answer

If you mean $\large 4^i$, then yes

anonymous · Answer

yes yes

kirbykirby · Answer

so the same can be done on the second term. Now you have 2 terms in the $\large\Sigma$. See if you can manipulate it so that you get only one sum with one term only, and verify if this has the form above, or can be manipulated as such.

anonymous · Answer

alright..

kirbykirby · Answer

So, if you cannot combine the terms, then you might not be facing a geometric sum. What you have is $$ \large \sum_{i=1}^n(4^i+8^i)=\sum_{i=1}^n\left(4(4^{i-1})+8(8^{i-1}\right) $$
Then you can start plugging in $i=1,2,3...$ and see if you observe the pattern:
$a+ar+ar^2+...$

Hint: the term at $i=1$ will be $a$, and then you can find the common ratio by doing $$ \frac{ar^2}{ar^1}=r$$ and see if you observe this for other numbers.