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How do you find the sum of the geometric series $4096 – 512 + 64 - …. to\,\, 5$ terms.

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Last updated date: 20th Jun 2024
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Answer
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Hint: Here in this question, we have to find the sum of finite geometric series. The geometric series is defined as the series with a constant ratio between the two successive terms. Then by considering the geometric series we have found the sum of the series.

Complete step-by-step solution:
In mathematics we have three types of series namely, arithmetic series, geometric series and harmonic series. The geometric series is defined as the series with a constant ratio between the two successive terms. The finite geometric series is generally represented as \[a,ar,a{r^2},...,a{r^n}\], where a is first term and r is a common ratio.
Now consider the series $4096 – 512 + 64 - …..$
Here the term a is known as first term. the value of a is 4096.
The r is the common ratio of the series. It is defined as \[r = \dfrac{{{a_2}}}{{{a_1}}}\]
The value of r is determined by \[r = \dfrac{{ - 512}}{{4096}} = - \dfrac{1}{8}\]
Now we have to find the sum of finite geometric series, the sum for finite geometric series is defined by \[{S_n}\]
Here the value of r is less than 1 we have a formula for the sum of geometric series and it is defined as
\[{S_n} = \dfrac{{a(1 - {r^n})}}{{(1 - r)}}\]
Here the value of n is 5.
Therefore by substituting the values in the formula we have
\[{S_5} = \dfrac{{4096\left( {1 - {{\left( {\dfrac{{ - 1}}{8}} \right)}^5}} \right)}}{{1 - \left( {\dfrac{{ - 1}}{8}} \right)}}\]
On simplifying we have
\[{S_5} = \dfrac{{4096\left( {1 + {{\left( {\dfrac{1}{8}} \right)}^5}} \right)}}{{1 + \dfrac{1}{8}}}\]
\[
   \Rightarrow {S_5} = \dfrac{{4096\left( {1 + \dfrac{1}{{32768}}} \right)}}{{\dfrac{{8 + 1}}{8}}} \\
   \Rightarrow {S_5} = \dfrac{{4096\left( {\dfrac{{32768 + 1}}{{32768}}} \right)}}{{\dfrac{{8 + 1}}{8}}} \\
   \Rightarrow {S_5} = \dfrac{{4096\left( {\dfrac{{32769}}{{32768}}} \right)}}{{\dfrac{9}{8}}} \\
   \Rightarrow {S_5} = \dfrac{{4096 \times 32769}}{{32768}} \times \dfrac{8}{9} \\
   \Rightarrow {S_5} = 3641 \\
 \]

Hence the sum of geometric series $4096 – 512 + 64 - … to\, 5$ terms is 3641.

Note: Three different forms of series are arithmetic series, geometric series and harmonic series. For the arithmetic series is the series with common differences. The geometric series is the series with a common ratio. The sum is known as the total value of the given series.