Answer

Verified

382.5k+ views

**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.

Recently Updated Pages

How many sigma and pi bonds are present in HCequiv class 11 chemistry CBSE

Why Are Noble Gases NonReactive class 11 chemistry CBSE

Let X and Y be the sets of all positive divisors of class 11 maths CBSE

Let x and y be 2 real numbers which satisfy the equations class 11 maths CBSE

Let x 4log 2sqrt 9k 1 + 7 and y dfrac132log 2sqrt5 class 11 maths CBSE

Let x22ax+b20 and x22bx+a20 be two equations Then the class 11 maths CBSE

Trending doubts

Which are the Top 10 Largest Countries of the World?

How many crores make 10 million class 7 maths CBSE

Fill the blanks with the suitable prepositions 1 The class 9 english CBSE

Difference between Prokaryotic cell and Eukaryotic class 11 biology CBSE

Change the following sentences into negative and interrogative class 10 english CBSE

Fill the blanks with proper collective nouns 1 A of class 10 english CBSE

Give 10 examples for herbs , shrubs , climbers , creepers

Fill in the blanks A 1 lakh ten thousand B 1 million class 9 maths CBSE

The Equation xxx + 2 is Satisfied when x is Equal to Class 10 Maths