Question

How do you write a rule for the ${{n}^{th}}$ term of the arithmetic sequence given $d=-3$ and ${{a}_{2}}=18$ ?

Answer 1

Hint: We start solving the problem by writing the general expression for the ${{n}^{th}}$ term of a arithmetic sequence is $a+\left( n-1 \right)d$ where a is the first term of the sequence and d is the constant common difference. We first put $n=2$ and $d=-3$ equate it to $18$ . From this, we get the value of a, which can be used to rewrite the general expression for the ${{n}^{th}}$ term.

Complete step by step answer:
Sequence is an enumerated collection of objects in which repetitions are allowed and where the order of objects is the most important. By objects, we mean generally numbers. Sequence may follow a specific pattern or can even be completely random. If the sequence follows a certain pattern, then the sequence can be of various types like arithmetic sequence, geometric sequence and so on.
The general expression for the ${{n}^{th}}$ term of a geometric sequence is $a+\left( n-1 \right)d$ where a is the first term of the sequence and d is the constant common difference between the consecutive terms. In the given problem, we are given the second term of the sequence and the value of the common difference. So, using the general expression for the ${{n}^{th}}$ term and putting the value of n equal to $2$ , we get,
$\begin{align}
  & \Rightarrow {{a}_{2}}=a+\left( 2-1 \right)\left( -3 \right)=18 \\
 & \Rightarrow a-3=18 \\
 & \Rightarrow a=21 \\
\end{align}$
So, the general rule thus becomes,
$\begin{align}
  & 21+\left( n-1 \right)\left( -3 \right) \\
 & =21+3-3n \\
 & =24-3n \\
\end{align}$

Therefore, we can conclude that the general expression for the ${{n}^{th}}$ term of the arithmetic sequence given $d=-3$ and ${{a}_{2}}=18$ is $24-3n$.

Note: In order to solve these problems, we must have some basic knowledge of the various types of sequences. We must be careful with the general expression of the ${{n}^{th}}$ term which is $a+\left( n-1 \right)d$ . Students often misinterpret it as $a+nd$ which leads to wrong answers. Also, sometimes the common difference can be negative as well such as in this very problem. We should not get confused as an arithmetic sequence is not bound to be increasing.