Question

If the sum of first n numbers of an AP \[2,4,6...\]is \[240\], then the value of n is
1. \[14\]
2. \[15\]
3. \[16\]
4. \[17\]

Answer 1

Hint: Now we know the basic properties of any arithmetic progression. First start by considering the first term of an arithmetic progression to be a and then consider the common difference between any two consecutive numbers in an arithmetic progression to be d. We must know the formula that the sum of the arithmetic equation is \[S=\dfrac{n}{2}(2a+(n-1)d)\].

Complete step-by-step solution:
Before we proceed with the solution here we need to start by recognizing which formula will be required to use here . In this question we are given the first term of an Arithmetic progression. We can find the common difference that is d in between any two consecutive numbers of the Arithmetic progression. And lastly we also know the sum of first n numbers of the arithmetic progression seeing all these we can say that we need to use the formula of sum of an AP which is
\[S=\dfrac{n}{2}(2a+(n-1)d)\]
Now here since the arithmetic progression is given to us which is \[2,4,6..\], we can say that the value of a here is
The first term of arithmetic progression (a) ;\[a=2\]
Now the common difference of the arithmetic equation given to us here is (d) ; \[d=4-2=2\]
Now the sum of first n numbers as given in the question is (S); \[S=240\]
Substituting the values we get
\[240=\dfrac{n}{2}(2(2)+(n-1)2)\]
Multiplying and solving the brackets
\[240=\dfrac{n}{2}(4+2n-2)\]
Simplifying and dividing we get
\[240=n(n+1)\]
We can write Sum as
\[15\times (15+1)=n(n+1)\]
Therefore we can find the value of n here which is \[n=15\]
Therefore if the sum of first n numbers of an AP \[2,4,6...\]is \[240\], then the value of n is \[15\].

Note: Arithmetic progression is a sequence of numbers where every successive term is a sum of its preceding term and a fixed difference. A common alternative here is you can find the nth term in this series and then find the value of n. The problem here is there can be many values in between making it difficult for us, So we prefer using this formula.