Question

How do you write the explicit rule for the ${{n}^{th}}$ term of the sequence $-1,2,5,8$ ?

Answer 1

Hint: For these kinds of questions, we should be aware of the concept of Arithmetic Progression. Arithmetic Progression or AP is a sequence of numbers such that the difference between the consecutive terms is constant. The first term is usually denoted by $a$. And the difference between the consecutive terms is called the common difference. It is usually denoted by $d$ . Here we are asked to write a rule to find out the ${{n}^{th}}$term. But we can rather find out the values of $a,d$ from the above sequence and substitute in the formula of ${{n}^{th}}$ term of an AP.

Complete step by step solution:
We are given the sequence of $-1,2,5,8$. Before actually proceeding further, let us check if this particular sequence satisfies the conditions of being an AP or not. For that, we have to check whether the difference between the consecutive terms is constant or not.
Our first term is $-1$ .
Let us subtract the consecutive terms and find out.
Upon doing so, we get the following :
$\begin{align}
  & \Rightarrow 2-\left( -1 \right)=3 \\
 & \Rightarrow 5-2=3 \\
 & \Rightarrow 8-5=3 \\
\end{align}$
It is a constant difference. This is an AP with the first term,$a$, to be $-1$ and the common difference,$d$, to be $3$. This is an increasing AP with where we add $3$ to every number.
There is already an established formula to find out the ${{n}^{th}}$term in an AP. It is as follows :
$\Rightarrow {{a}_{n}}=a+\left( n-1 \right)d$, where ${{a}_{n}}$ denoted the ${{n}^{th}}$term and $n$ denoted the number of terms.
Since we are asked to find out ${{n}^{th}}$term, our number of terms i.e $n$ which is to be substituted in the formula should also be $n$. If we are asked a specific number such ${{7}^{th}}$ term of this AP, then our $n$ would be $7$.
Since we know all our variables, let us substitute them and find the rule to find out our ${{n}^{th}}$term.
Upon substituting , we get the following :
$\begin{align}
  & \Rightarrow {{a}_{n}}=a+\left( n-1 \right)d \\
 & \Rightarrow {{a}_{n}}=\left( -1 \right)+\left( n-1 \right)3 \\
 & \Rightarrow {{a}_{n}}=-1+3n-3 \\
 & \Rightarrow {{a}_{n}}=3n-4 \\
\end{align}$
$\therefore $ Hence, the explicit rule for the ${{n}^{th}}$ term of the sequence $-1,2,5,8$ is ${{a}_{n}}=3n-4$.

Note: Using the formulae of AP decreases our time and we will end up with the answer quickly. So we should remember all the formulae regarding Arithmetic Progression and also some of it’s short-cuts. If we are not confident with our rule in this question, we can substitute the value of $n$ in the rule we get. Upon substituting $n=1,2,3..$ , we will end up with the terms of the sequence which are mentioned in the question. We have to be careful while doing calculations as there is a lot of scope of calculation errors.