MHB Need Sum of Formula [shortcut]

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The discussion focuses on solving a problem involving the sum of a geometric series. The formula for the sum of the first n terms of a geometric series is provided, which is S_n = a(r^n - 1) / (r - 1). Two specific examples of geometric series are presented: 1 + 7^1 + 7^2 + 7^3 + 7^4 + 7^5 and 3^1 + 3^2 + 3^3 + 3^4 + 3^5 + 3^6 + 3^7. Participants are encouraged to apply the formula to these examples to find the respective sums. The conversation emphasizes the importance of understanding geometric series for solving related problems.
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hi guys.

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

please, see attachment file..

thanks for your helping..

susanto3311
 

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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)?
 
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: $\:\begin{Bmatrix}a &=& \text{first term} \\ r &=& \text{common ratio} \\ n &=& \text{no. of terms}\end{Bmatrix}$
 
Seemingly by some mathematical coincidence, a hexagon of sides 2,2,7,7, 11, and 11 can be inscribed in a circle of radius 7. The other day I saw a math problem on line, which they said came from a Polish Olympiad, where you compute the length x of the 3rd side which is the same as the radius, so that the sides of length 2,x, and 11 are inscribed on the arc of a semi-circle. The law of cosines applied twice gives the answer for x of exactly 7, but the arithmetic is so complex that the...

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