Question

What is the correct formula to find the sum of the finite geometric series below? $\displaystyle 2 + \frac{2}{3} + \frac{2}{9} + \ ... \ + \frac{2}{3^6}$​

Answer 1

$\boxed{\boxed{\huge{\red{\bold{\underline{Answer:-}}}}}}$

We are given the Geometric series:

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

which can be written as:

$2 + \frac{2}{3} + \frac{2}{3 {}^{2} } + \frac{2}{3 {}^{3} } + \frac{2}{3 {}^{4} } + \frac{2}{3 {}^{5} } + \frac{2}{3 {}^{6} }$

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

Hence, we can see that the common ratio of this geometric series is 1/3.

Finding the sum:

We know that the sum of a geometric series is:

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

(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 after terms is 7.

plugging these values in the formula, we get

$Sn = \frac{2((1/3) { }^{7} - 1)}{(1/3) - 1}$

$Sn = \frac{ - 1.99}{ - 0.67}$

$\fbox\pink{Sn=2.97}$

Answer 2

$\large\underline{\sf{Solution-}}$

Wᴇ ᴋɴᴏᴡ ᴛʜᴀᴛ,

↝ Sum of infinite geometric sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_ \infty \:=\dfrac{a}{1 \: - \: r} \: provided \: that \: |r| < 1}}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

a is the first term of the sequence.

r is the common ratio.

Now, given infinite series is

$\red{\rm :\longmapsto\:2 + \dfrac{2}{3} + \dfrac{2}{9} + \dfrac{2}{27} + - - - \infty }$

Since,

Ratio between the consecutive terms remains the same, so given series is infinite GP series with

$\rm :\longmapsto\:First \: term, \: a = 2$

and

$\rm :\longmapsto\:Common \: ratio, \: r = \dfrac{1}{3}$

So,

Sum of this infinite GP series is given by

$\rm :\longmapsto\:S_ \infty = \dfrac{a}{1 - r}$

On substituting the values of a and r, we get

$\rm :\longmapsto\:S_ \infty = \dfrac{2}{1 - \dfrac{1}{3} }$

$\rm :\longmapsto\:S_ \infty = \dfrac{2}{\dfrac{3 - 1}{3} }$

$\rm :\longmapsto\:S_ \infty = \dfrac{2}{\dfrac{2}{3} }$

$\rm :\longmapsto\:S_ \infty = 2 \times \dfrac{3}{2}$

$\bf:\longmapsto\:S_ \infty = 3$

Additional Information :-

↝ Sum of n  terms of an geometric sequence is,

$\begin{gathered}\red\bigstar\:\:{\underline{\orange{\boxed{\bf{\green{S_n\:=\dfrac{a \: ( \: {r}^{n} \: - \: 1 \: ) }{r \: - \: 1} \: where \: r \: \ne \: 1 }}}}}} \\ \end{gathered}$

Wʜᴇʀᴇ,

Sₙ is the sum of n terms of GP.

a is the first term of the sequence.

n is the no. of terms.

r is the common ratio.

↝ If a, b, c are in GP then

$\boxed{ \bf{ \: {b}^{2} \: = \: ac \: }}$