Question

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

Answer 1

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.

{{{\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 $80$