Question

The first term of an AP is 5, the common difference is 3 and the last term is 80. Find the number of terms.

Answer 1

Hint: We solve this problem by using the ${{n}^{th}}$ term of an AP.
If $'a'$ is the first term and $'d'$ is the common difference of an AP then the ${{n}^{th}}$ term of the AP is given as,
${{T}_{n}}=a+\left( n-1 \right)d$
By using this formula to the given data we find the required number of terms by using the condition that the value of $'n'$ for the last term will be the number of terms of given AP.

Complete step by step solution:
We are given that the first term of the AP as 5
Let us assume that the first term of the AP as,
$\Rightarrow a=5$
We are also given the common difference of the AP as 3.
Let us assume that the given common difference of the AP as,
$\Rightarrow d=3$
We are asked to find the total number of terms in the AP if the last term is 80.
Let us assume that there are $'n'$ number of terms in the AP and assume that last term as ${{n}^{th}}$ term of AP that is,
$\Rightarrow {{T}_{n}}=80$
Now, let us use the formula of ${{n}^{th}}$ term of the AP.
We know that the ${{n}^{th}}$ term of the AP is given as,
${{T}_{n}}=a+\left( n-1 \right)d$
Where, $'a'$ is the first term and $'d'$ is the common difference of an AP.
By using this formula for the given data then we get,
$\begin{align}
  & \Rightarrow 80=5+\left( n-1 \right)\left( 3 \right) \\
 & \Rightarrow 3n-3=75 \\
 & \Rightarrow 3n=78 \\
\end{align}$
By dividing the both sides of the equation with 3 we get,
$\begin{align}
  & \Rightarrow \dfrac{3n}{3}=\dfrac{78}{3} \\
 & \Rightarrow n=26 \\
\end{align}$

Therefore, we can conclude that there are a total of 26 terms in the given AP.

Note: Students may make a mistake by adding ‘1’ to the value of $'n'$ because they misunderstand that the first term is not counted. In the formula of ${{T}_{n}}$ the value $'n'$ represents the number of the term.
If $n=1$ then it says that ${{T}_{1}}$ which is the first term of AP. So, if the last term has $n=26$ then we can directly say that there are 26 terms. There will be no need to add ‘1’.