Question

What is the $n^{th}$ term of the arithmetic sequence 1,9,17,25?

Answer 1

The equation to find the $n^{th}$ term or a general term of an arithmetic sequence is given as ${{t}_{n}}=a+\left( n-1 \right)d$ , where ${{t}_{n}}$ is the $n^{th}$ term of the sequence, a is the first term of the sequence, n is the number of terms and d is the common difference of the sequence. We can use this equation to find the $n^{th}$ term of the given arithmetic sequence.

Complete step by step answer:
An arithmetic sequence is defined as one whose successive terms have a common difference. The given sequence in the question is 1,9,17,25. We can use the formula for the $n^{th}$ term of an arithmetic sequence to solve this question. It is given as ${{t}_{n}}=a+\left( n-1 \right)d$.
From the sequence given, we can clearly see that the first term a is equal to 1. The common difference can be found as the difference of any two successive terms. Taking the first two terms, we get the common difference as,
$\begin{aligned} & \,\,\,\,\,\,d=9-1 \\\ & \Rightarrow d=8 \\\ \end{aligned}$
Finally, substituting these values in the equation for $n^{th}$ term of an arithmetic sequence, we get,
$\begin{aligned} & \,\,\,\,\,\,{{t}_{n}}=a+\left( n-1 \right)d \\\ & \,\,\,\,\,\,{{t}_{n}}=1+\left( n-1 \right)8 \\\ & \Rightarrow {{t}_{n}}=1+8n-8 \\\ & \Rightarrow {{t}_{n}}=8n-7 \\\ \end{aligned}$

Therefore, the $n^{th}$ term of the given sequence is given by ${{t}_{n}}=8n-7$.

Note: After finding the $n^{th}$ term of the sequence, we can cross check or verify if the expression is correct by substituting n starting from 1 and comparing them with the given sequence. For example, let us substitute n as 1 in the expression we have found, we get,
$\begin{aligned} & {{t}_{1}}=8\left( 1 \right)-7 \\\ & {{t}_{1}}=1 \\\ \end{aligned}$
Hence this matches with the sequence. For $n=2$ , we get,
$\begin{aligned} & {{t}_{2}}=8\left( 2 \right)-7 \\\ & {{t}_{2}}=9 \\\ \end{aligned}$
Even this value agrees with the given sequence. For $n=3$ , we get,
$\begin{aligned} & {{t}_{3}}=8\left( 3 \right)-7 \\\ & {{t}_{3}}=17 \\\ \end{aligned}$
Even this value agrees with the given sequence. Finally, for $n=4$ , we get,
$\begin{aligned} & {{t}_{4}}=8\left( 4 \right)-7 \\\ & {{t}_{4}}=25 \\\ \end{aligned}$
Therefore, all the values match with the given sequence and the expression for the $n^{th}$ term of the sequence is verified to be correct.