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The sum of the first n term of the sequence is given as \[{{2}^{n}}+{{2}^{n-1}}\], how do you work out the \[{{10}^{th}}\] term of the sequence?

seo-qna
Last updated date: 22nd Feb 2024
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IVSAT 2024
Answer
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Hint: This is a question of Series and sequences. In sequences, where the sum of n terms is given, to calculate a particular term we simply find the sum of n terms from the given formula by putting the value of n as given and then we find the sum of (n-1) terms. The nth term value is given as the difference between the sum of n terms and the sum of (n-1) terms.

Complete step by step answer:
Here the nth term we need to find is \[{{10}^{th}}\] term
Now let us name the sum of the first n terms of a sequence as \[{{S}_{n}}\]
thus, \[{{S}_{n}}={{2}^{n}}+{{2}^{n-1}}.........(1)\]
Now we simply substitute the value of n= 10 in the above expression to get the sum of the first 10 terms.
We get,
\[{{S}_{10}}={{2}^{10}}+{{2}^{10-1}}\]
\[\Rightarrow {{2}^{10}}+{{2}^{9}}\]
Simplifying further, taking \[{{2}^{9}}\] common we get
\[{{2}^{9}}(2+1)\]
\[\Rightarrow {{2}^{9}}(3)\]
Hence the sum of the first 10 terms is given as \[3({{2}^{9}})\]
You may also expand the solution by putting the value of \[{{2}^{9}}\] as 512
Hence the sum of the first 10 terms is \[512\times 3=1536\]
\[{{S}_{10}}=1536\]
Now we will find the sum of the first (n-1) terms
As you know n=10 hence (n-1) is given as
 (10-1), that is 9
Substituting this value in expression (1) to find the sum of the first 9 terms.
We get
\[{{S}_{n}}={{2}^{n}}+{{2}^{n-1}}\]
\[{{S}_{9}}={{2}^{9}}+{{2}^{9-1}}\]
\[\Rightarrow {{2}^{9}}+{{2}^{8}}\]
We will be taking \[{{2}^{8}}\] as common to simplify the expression
\[{{2}^{8}}(2+1)\]
\[\Rightarrow {{2}^{8}}(3)\]
Hence the sum of the first 9 terms is given as
\[{{S}_{9}}={{2}^{8}}(3)\]
Expand the term \[{{2}^{8}}\], we will get
\[{{S}_{9}}=256(3)\] or,
\[{{S}_{9}}=768\]
Now here we have calculated the sum of the first 10 terms and 9 terms.
The sum of the first 10 terms can also be given as the sum of the first 9 terms + \[{{10}^{th}}\] term.
\[{{S}_{10}}={{10}^{th}}Term+{{S}_{9}}\]
Reshifting the terms we get
\[\Rightarrow {{S}_{10}}-{{S}_{9}}={{10}^{th}}Term\]
Putting values of \[{{S}_{10}}\] and \[{{S}_{9}}\] as calculated above we get
\[\Rightarrow 1536-768={{10}^{th}}Term\]
\[\Rightarrow 768={{10}^{th}}Term\]
Hence the \[{{10}^{th}}\] Term of the sequence is given as 768.

Note:
In some questions, the nth term of the sequence is not the same as the term number, which is sometimes in sequences such as S = 0, 1, 2, 3, 4…. The \[{{4}^{th}}\] term is not 4 whereas it is 3, that is (n-1) th term.
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