Question

The sum of n terms of the series 2, 5, 8, ………. is 950. Find n.

Answer 1

Hint: We will first of all mention the formula used for the sum of n terms in an A. P. and then substitute it equal to 950. Then do the required modifications and thus we have the answer.

Complete step-by-step solution:
We are given an arithmetic progression: 2, 5, 8, ……….
Here, we can clearly see that the first term = a = 2
We know that the common difference is the difference of a succeeding term and the term itself.
Here, common difference = 5 – 2 = 3
So, we get d = 3.
Now, we also know that the sum of n terms in an A. P. is given by the following formula:-
$ \Rightarrow {S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$
Putting the values a = 2 and d = 3 and the given sum as 950 to get the following expression:-
$ \Rightarrow 950 = \dfrac{n}{2}\left[ {2 \times 2 + \left( {n - 1} \right) \times 3} \right]$
Simplifying the calculations inside the parenthesis on the right hand side to obtain the following expression:-
$ \Rightarrow 950 = \dfrac{n}{2}\left[ {4 + 3\left( {n - 1} \right)} \right]$
Simplifying the calculations further by eradicating the [ ] bracket from the right hand side in the above expression to get the following expression:-
$ \Rightarrow 950 = 2n + \dfrac{{3n(n - 1)}}{2}$
Multiplying by 2 the whole equation to get the following expression:-
$ \Rightarrow 1900 = 4n + 3n(n - 1)$
Simplifying the calculations further by eradicating the ( ) bracket from the right hand side in the above expression to get the following expression:-
$ \Rightarrow 1900 = 4n + 3{n^2} - 3n$
By simplifying the above expression, we will get the following expression:-
$ \Rightarrow 3{n^2} + n - 1900 = 0$
We know that for any general equation: $a{x^2} + bx + c = 0$, the roots of the equation are given by:
$ \Rightarrow x = \dfrac{{ - b \pm \sqrt {{b^2} - 4ac} }}{{2a}}$
Therefore, the roots of $3{n^2} + n - 1900 = 0$ is:
$ \Rightarrow n = \dfrac{{ - 1 \pm \sqrt {1 + 22800} }}{6}$
Simplifying the calculations inside the square root:-
$ \Rightarrow n = \dfrac{{ - 1 \pm \sqrt {22801} }}{6}$
Simplifying the calculations further to get the following expression:-
$ \Rightarrow n = \dfrac{{ - 1 \pm 151}}{6}$
On solving the above expression for both the possible values, we will get:-
$ \Rightarrow n = \dfrac{{150}}{6}$ and $n = \dfrac{{152}}{6}$.
Simplifying the calculations to get the required answer:-
$ \Rightarrow n = 25$ and $n = \dfrac{{76}}{3}$.
Since the number of terms cannot be in fraction, therefore, n = 25.

Hence, the value of n is 25.

Note: The students must commit to the memory the following formula:-
$ \Rightarrow {S_n} = \dfrac{n}{2}\left[ {2a + \left( {n - 1} \right)d} \right]$, where n is the number of terms, a is the first term and d is the common difference of the given arithmetic progression.
The students must also note that they should find both the possible solutions by using the formula for finding roots and then discard one value according to the given data only and if nothing can be discarded, that means there are two possible values of n, we can take.