# 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.

Last updated date: 25th Mar 2023

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Answer

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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.

Complete step-by-step answer:

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, \[{S_n} = \dfrac{n}{2}\left( {a + l} \right)\] .

Complete step-by-step answer:

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, \[{S_n} = \dfrac{n}{2}\left( {a + l} \right)\] .

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