Question

# The first term of the A.P is 5, the last term is 45 and the sum is 400. Find the number of terms in the A.P.

Hint: We can find the number of terms in the Arithmetic Progression (A.P) by using the formula for sum of $n$ terms of the A.P where the ${n^{th}}$ term is given as 45 and the sum is 400.

Arithmetic Progression (A.P) is a sequence whose terms increase or decrease by a fixed number called the common difference.
If a is the first term of the A.P and d is the common difference of the A.P, then $l$ , the ${n^{th}}$ term of the A.P is given as follows:
$l = a + (n - 1)d{\text{ }}..........{\text{(1)}}$
The sum of n terms of the A.P, ${S_n}$ is given by:
${S_n} = \dfrac{n}{2}\left[ {2a + (n - 1)d} \right]{\text{ }}...........{\text{(2)}}$
Equation (2) can be written in terms of $l$ , the ${n^{th}}$ term by using equation (1).
${S_n} = \dfrac{n}{2}\left( {a + l} \right){\text{ }}..........(3)$
The value of first term, the last term and the sum to n terms of the A.P is given.
$a = 5$
$l = 45$
${S_n} = 400$
Using these values in equation (3) and solving for $n$ , we get:
$400 = \dfrac{n}{2}\left( {5 + 45} \right)$
Simplifying the RHS, we get:
$400 = \dfrac{n}{2}\left( {50} \right)$
Dividing 50 by 2 we get 25, hence, we have:
$400 = 25n$
Solving for n by dividing 400 by 25, we get:
$n = \dfrac{{400}}{{25}}$
$n = 16$
Hence, the number of terms of the A.P is 16.

Note: You can not solve the equation by just using the first and the last term using the formula for the ${n^{th}}$ term of the A.P since the common difference is not given. Also, you must know the second form of sum to n terms of A.P, that is, ${S_n} = \dfrac{n}{2}\left( {a + l} \right)$ .