How To Write A Formula For The First N Terms of a Sequence?

Question

So I have the sequence $$\{5, 15, 45,\cdots\}$$ and I figured out that the formula to find a particular term is $$S_n = 5 \times 3^{n-1}$$ but how do I use this to find the sum of the first $n$ terms?

Answer 1

Hint. One may recall that, for $n=1,2,\cdots$, $$ 1+x+x^2+\cdots+x^{n-1}=\frac{1-x^{n}}{1-x}, \qquad x \neq 1. $$

Answer 2

Geometric Summation: Given the sequence$$a,ar,ar^2,ar^3,\ldots,ar^{n-1}\tag1$$ We have the sum as$$S=a\left(\frac {1-r^n}{1-r}\right)\tag2$$ Where $S$ is the total sum and $a$ is the first term of the sequence.

Proof: Multiplying $S$ by $r$, we obtain another equation$$Sr=ar+ar^2+ar^3+\ldots+ar^n\tag3$$ And subtracting $Sr$ from $S$, we have$$S-Sr=a-ar^n=a\left(1-r^n\right)\implies \boxed{S=a\frac {1-r^n}{1-r}}\tag4$$

For example, if you wanted to find the sum of the first two terms $5+15$, we see that $a=5$, $r=3$ and $n=2$. Thus, the sum is$$S=5\cdot\frac {1-3^2}{1-3}=20$$ Trying out the first $5$ sums, we have $5+15+45+135+405$. Therefore,$$S=5\cdot\frac {1-3^5}{1-3}=605$$ And verifying, we see that indeed, the sum is $605$.$$5+15+45+135+405=605$$