Question

# The first term of an AP is 5 and the last term is 45. If the sum of the terms of the AP is 120, then find the number of terms.

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Hint: In order to solve this problem, we need to know the meaning of the term arithmetic progression. An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant. The formula for finding the sum of n terms is called $Sum=\dfrac{n}{2}\left( a+l \right)$, where, n is the number of terms, a is the first term in the series, l is the last term in the series.

Complete step-by-step solution:
In this question, we are given the sum of a certain number of terms in arithmetic progression.
Lets first understand by the term arithmetic progression.
An arithmetic progression is a sequence of numbers such that the difference of any two successive members is a constant.
Also, we are given the first term of the sequence.
The first term is 5.
And the last term given is 45.
We need to know the formula for the sum of n terms.
The formula is given as follows.
$Sum=\dfrac{n}{2}\left( a+l \right)$
Where n is the number of terms,
a is the first term in the series,
l is the last term in the series.
Now, according to the condition we just have one unknown that is the number of terms.
Therefore, substituting the values a = 5, l = 45, Sum = 120,
We get,
$120=\dfrac{n}{2}\left( 5+45 \right)$
Solving the equation, we get,
\begin{align} & 120=\dfrac{n}{2}\left( 50 \right) \\ & n=\dfrac{120\times 2}{50} \\ & n=4.8 \\ \end{align}
n is bounded to be an integer. Hence, this series is not possible.

Note: In this problem, we need to use the formula which has the last term and the first term in it. It is often misunderstood. We can solve this by another approach. The formula is given by $sum=\dfrac{n}{2}\left( 2a+\left( n-1 \right)d \right)$ , where d is the difference between the two consecutive terms. But now there are two unknowns, so we need to find another equation and the whole question becomes complicated.