Question

What is the correct formula to find the sum of the finite geometric series below?

$\sf 2 + \dfrac{2}{3} + \dfrac{2}{9} + \ ... \ + \dfrac{2}{3^6}$


ONLY QUALITY ANSWERS NEEDED ​

Answer 1

Answer:

$\sf 2 + \dfrac{2}{3} + \dfrac{2}{9} + \ ... \ + \dfrac{2}{3^6}$

To find the sum of a finite geometric series, use the formula, Sn=a1(1−rn)/1−r,r≠1 , where n is the number of terms, a1 is the first term and r is the common ratio .

Answer 2

$\huge\qquad \mathbb{\fcolorbox{red}{lavenderblush}{✰Answer}}$

We are given the Geometric Series:

$\sf {2 + \dfrac{2}{3} + \dfrac{2}{9} + \dfrac{2}{27} + \dfrac{2}{81} + \dfrac{2}{243} + \dfrac{2}{729}}$

which can be rewritten as:

$\tt{ 2 + \dfrac{2}{3} + \dfrac{2}{3^{2} } + \dfrac{2}{3^{3}} + \dfrac{2}{3^{4}} + \dfrac{2}{3^{5}} + \dfrac{2}{3^{6}}}$

here, we can see that every term is (1/3) times the last term

Hence, we can say that the common ratio of this Geometric Series is 1/3

Finding the Sum:

We know that the sum of a Geometric Series is:

$\boxed{\bold {S_{n} = \dfrac{a(r^{n}-1)}{r-1}}}$

(where r is the common ratio, a is the first term, and n is the number of terms)

another look at the given Geometric Series tells us that the first term is 2 and the number of terms is 7

plugging these values in the formula, we get:

$\sf{ S_{n} = \dfrac{2((1/3)^{7}-1)}{(1/3)-1}}$

$\sf {S_{n} = \dfrac{-1.99}{-0.67}}$

Sₙ = 2.97

 

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