MHB Need Sum of Formula [shortcut]

[susanto3311]

hi guys.

i have 2 questions, how do solve this problem with formula [shortcut] :

please, see attachment file..

thanks for your helping..

susanto3311

 

[MarkFL]

Consider the following sum:

$$S_n=a^0+a^1+a^2+\cdots+a^n\tag{1}$$

Now multiply through by $a$:

$$aS_n=a^1+a^2+a^3+\cdots+a^{n+1}\tag{2}$$

What do you get if you subtract (1) from (2)?

 

[soroban]

Hello, susanto3311!

$1 + 7^1 + 7^2 + 7^3 + 7^4 + 7^5 \:=\:? $

$3^1 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + 3^7 \:=\:?$

These are Geometric Series.
MarkFL indicated how we find the formulas for these series.

The sum of the first $n$ terms of Geometric Series

$\;\;\;$is given by: $\:S_n \;=\;a\,\dfrac{r^n\,-\,1}{r\,-\,1}$

where: $\:\begin{Bmatrix}a &=& \text{first term} \\ r &=& \text{common ratio} \\ n &=& \text{no. of terms}\end{Bmatrix}$