Question

Which term of the A.P $5,9,13,17...$ is \[81\] ?

Answer 1

Hint: An arithmetic sequence or arithmetic progression can be defined as a mathematical sequence and the difference between two consecutive terms is always a constant; an arithmetic progression is abbreviated as A.P.
Here, we are asked to calculate the ${n^{th}}$term of the given arithmetic progression.
And the given arithmetic progression is $5,9,13,17...$
Also, ${n^{th}}$term of the given arithmetic progression is \[81\].
To find the ${n^{th}}$term, we need to find the first term and the common difference of the given arithmetic progression and we need to substitute in the formula.

Formula used:
The formula to calculate the ${n^{th}}$term of the given arithmetic progression is as follows.
\[\begin{array}{*{20}{l}}
  {{{\mathbf{a}}_{\mathbf{n}}} = {\mathbf{a}} + \left( {{\mathbf{n}} - {\mathbf{1}}} \right) \times {\mathbf{d}}}
\end{array}\]
Where, $a$denotes the first term, $d$ denotes the common difference, $n$is the number of terms, and ${a_n}$is the ${n^{th}}$term of the given arithmetic progression.

Complete step-by-step answer:
The given A.P. is $5,9,13,17...$
Also, it is given that
${n^{th}}$term of the given arithmetic progression ${a_n} = 81$
Here, the first term \[a = 5\]
The common difference \[d = 9 - 5 = 4\]
To find: $n$
Now, we need to use the formula,
\[\begin{array}{*{20}{l}}
  {{{\mathbf{a}}_{\mathbf{n}}} = {\mathbf{a}} + \left( {{\mathbf{n}} - {\mathbf{1}}} \right) \times {\mathbf{d}}}
\end{array}\]
Now, we shall substitute the known values on the above formula, we get
\[81 = 5 + (n - 1) \times 4\]
\[ \Rightarrow 81 = 5 + 4n - 4\]
\[ \Rightarrow 81 = 1 + 4n\]
\[ \Rightarrow 81 - 1 = 4n\]
On solving it further, we have
\[ \Rightarrow 4n = 80\]
$ \Rightarrow n = \dfrac{{80}}{4}$
\[ \Rightarrow n = 20\]
Hence, the ${20^{th}}$term of the given arithmetic progression is $80$ is the required answer too.

Note: We also need to learn the three important terms, which are as follows.
A common difference $\left( d \right)$ is a difference between the two terms.
${n^{th}}$term \[({a_n})\]
And, Sum of the first $n$ terms \[({S_n})\]
Here, we were asked to calculate the ${n^{th}}$term of the given arithmetic progression.
Hence, the ${20^{th}}$term of the given arithmetic progression is $80$