Answer
384.6k+ views
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.
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.
Recently Updated Pages
How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Why Are Noble Gases NonReactive class 11 chemistry CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let X and Y be the sets of all positive divisors of class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Trending doubts
Fill the blanks with the suitable prepositions 1 The class 9 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
At which age domestication of animals started A Neolithic class 11 social science CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Which are the Top 10 Largest Countries of the World?
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Give 10 examples for herbs , shrubs , climbers , creepers
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Difference Between Plant Cell and Animal Cell
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Write a letter to the principal requesting him to grant class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Change the following sentences into negative and interrogative class 10 english CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)
Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE
![arrow-right](/cdn/images/seo-templates/arrow-right.png)