Question

How do you find a formula for the nth term of the arithmetic sequence $24,32,40,48.....$?

Answer 1

Hint: In the above question, we have been given an arithmetic sequence whose nth term is to be determined. To find the nth term, we have to use the fact that the nth term of an arithmetic sequence is a linear function of n, so that we can let the nth term of the given sequence as ${{a}_{n}}=bn+c$. Since in this function, we have two unknown variables, $b$ and $c$, we need two equations in terms of these two variables. The two equations can be obtained by substituting the first and the second terms from the given sequence into the assumed function ${{a}_{n}}=bn+c$ for $n=1$ and $n=2$. On solving the two equations obtained, we will get the values of $b$ and $c$, and hence the nth term.

Complete step by step solution:
The given arithmetic sequence in the above question is $24,32,40,48.....$
We know that the nth term of an arithmetic sequence is a linear function of n. So we can let the nth term of the given arithmetic sequence as
$\Rightarrow {{a}_{n}}=bn+c........\left( i \right)$
In the given sequence, the first term is equal to $24$. Therefore, on substituting $n=1$ and ${{a}_{1}}=24$ in (i) we get
$\begin{align}
  & \Rightarrow 24=b\left( 1 \right)+c \\
 & \Rightarrow b+c=24........\left( ii \right) \\
\end{align}$
Also, in the given sequence, the second term is equal to $32$. Therefore, we substitute $n=2$ and ${{a}_{2}}=32$ in the equation (i) to get
$\begin{align}
  & \Rightarrow 32=b\left( 2 \right)+c \\
 & \Rightarrow 2b+c=32.......\left( iii \right) \\
\end{align}$
Subtracting (iii) from (ii) we get
$\begin{align}
  & \Rightarrow 2b+c-\left( b+c \right)=32-24 \\
 & \Rightarrow 2b+c-b-c=8 \\
 & \Rightarrow b=8 \\
\end{align}$
Substituting this in the equation (ii) we get
$\begin{align}
  & \Rightarrow 8+c=24 \\
 & \Rightarrow c=24-8 \\
 & \Rightarrow c=16 \\
\end{align}$
Substituting the above two values in the equation (i) we finally obtain
$\Rightarrow {{a}_{n}}=8n+16$
Hence, the nth term for the given arithmetic sequence is ${{a}_{n}}=8n+16$.

Note:
We can also use the general term of an AP to find the formula for the nth term of the given sequence, which is given by ${{a}_{n}}=a+\left( n-1 \right)d$. Here $a$ and $d$ are the first term and the common difference of the arithmetic sequence. The coefficient of $n$ in the nth term of an AP is equal to the common difference. Therefore, the value of $b$ in the assumed formula ${{a}_{n}}=bn+c$ can be directly substituted as the common difference of the given AP.