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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?

Last updated date: 13th Jul 2024
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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.

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.