Question

The number of terms in the AP 2,5,8, …, 59 are?

Answer 1

Hint: In the given question we are given an arithmetic progression in which we are provided with the first term, some middle terms and the last term of the progression. Now, we are being asked to find the total number of numbers that the given arithmetic progression contains.

Complete step-by-step solution:
According to the given question, we need to find the total number of terms present in the arithmetic progression. We know that the first term of the given progression is 2 and the next consecutive term is 5. So, now we can find the common difference of the given AP using these two terms which is 3.
Now, we are also given the last term of the AP which is 59 and we know the formula used to find the last terms of the AP which is ${{a}_{n}}= a+\left( n-1 \right)d$ , where a is the first term, d is common difference and n is the total number of terms of the AP and we know the value of ${{a}_{n}}$ which is the last term 59.
Now, substituting all these values in the formula we will get the value of n.
\[\begin{align}
  & {{a}_{n}}= a+\left( n-1 \right)d \\
 & \Rightarrow 59=2+\left( n-1 \right)3 \\
 & \Rightarrow 57=\left( n-1 \right)3 \\
 & \Rightarrow n-1=\dfrac{57}{3} \\
\end{align}\]
Now, simplifying this we will get the value of n as:
$\begin{align}
  & n=\dfrac{60}{3} \\
 & \Rightarrow n=20 \\
\end{align}$
Therefore, the total number of terms present in the given arithmetic progression is 20.

Note: In the question, we have directly substituted the value in the formula in order to cross verify we can find the common difference from the first two terms and then we can reach till last term but this is time consuming if we have a large number of terms. Therefore, we need to use the formula and be careful during the calculations.