Sum of N Terms

The series known as arithmetic progression is one in which the difference between any two subsequent numbers is kept constant. It is also called Arithmetic Sequence. For instance, the natural number sequence 1, 2, 3, 4, 5, 6, ... is an arithmetic progression with a common difference of 1 between any two succeeding terms. In this article, we will learn what is arithmetic progression, summation arithmetic progression, and formulas that can be used to do the calculation as well as solve examples for better understanding.

Sum of First N Terms Formula

The sum of n terms of an AP can be easily found using a simple formula that says that, if we have an AP whose first term is a and d is a common difference, then the formula for the sum of n terms of the arithmetic progression is \[{S_n} = \left( {\dfrac{n}{2}} \right)[2a + (n - 1)d]\].

To put it another way, the following equation can be used to determine the sum of the first n terms of an AP when it is given in the form of "a, a + d, a + 2d, a + 3d, ..., a + (n - 1) d":

Sum = \[\left( {\dfrac{n}{2}} \right)[2a + (n - 1)d]\]

Steps to Find the Sum

The sum of arithmetic progression of n terms requires the following steps:

Step 1: Find the first term of an Arithmetic Progression that is a

Step 2: Find the common difference between the two consecutive terms; we will get d

Step 3: Determine the n^th term

Step 4: Substitute a, d, n in the formula \[{S_n} = \left( {\dfrac{n}{2}} \right)[2a + (n - 1)d]\]

Shortcut Tips and Tricks

• When the nth term of an arithmetic progression i.e., an is given, we can use the formula $[{S_n} = \dfrac{n}{2}({a_1} + {a_n})\]$

Here a1 = first term and an = nth term

• \[\dfrac{{n(n + 1)}}{2}\] can be used to find the sum of natural numbers.

• \[\dfrac{{n(n + 1)(2n + 1)}}{6}\] can be used to find the sum of the square of n terms (ex: $1^2 + 2^2 + 3^2+ 4^2+ .. + n^2)$

• \[{\left[ {\dfrac{{n(n + 1)}}{2}} \right]^2}\]can be used to find the sum of the cube of n terms (ex:$1^3 + 2^3 + 3^3+ 4^3+ .. + n^3)$

Examples of Finding Sum in an AP

1. If the first term of an AP is 10 and the common difference is 2, find the sum of the first 10 terms.

Ans: Given, a = 10, n = 10, and d= 2

\[{S_n} = \left( {\dfrac{n}{2}} \right)[2a + (n - 1)d]\]

\[\begin{array}{l}{S_{10}} = \left( {\dfrac{{10}}{2}} \right)[2(10) + (10 - 1)2]\\{S_{10}} = 5[20 + 9(2)]\\{S_{10}} = 5[20 + 18]\\{S_{10}} = 5[38]\\{S_{10}} = 190\end{array}\]

Therefore, 190 is the sum.

2. Find the sum of the first 20 terms of the A.P.

1, 3, 5, 7, 9, 11………….

Ans: Given series 1, 3, 5, 7, 9, 11......

We got common difference (d) as (3 – 1) = (5 – 3) = ....... = 2

Given finding the sum of the first 20 terms

i.e., n = 20

Now, we got a = 1, d = 2, and n = 20

\[{S_n} = \left( {\dfrac{n}{2}} \right)[2a + (n - 1)d]\]

\[{S_{20}} = \left( {\dfrac{20}{2}} \right)[2 + (20 - 1)2]\]

\[{S_{20}}= 10[2(1)+(20-1)2]\]

\[{S_{20}}= 10[2+19(2)]\]

\[{S_{20}}= 10[40]\]

\[{S_{20}}= 400\]

Therefore, the sum of the first 20 terms of the series is 400.

3. Given a = 2, d = 2, and an = 20. Find the value of Sn.

Ans: The given values are a1 = 2 = a, d = 2, and a_n=20.

Formula for nth term = a+(n−1)d.

i.e., 20 = 2 +(n−1)2

20 = 2 + 2n−2

20 = 2n

n = 10

Finding the sum of AP when the nth term is given

\[\begin{array}{l}{S_n} = \dfrac{n}{2}(a + {a_n})\\{S_{10}} = \dfrac{{16}}{2}(2 + 20)\\{S_{10}} = 8(22)\\{S_{10}} = 176\end{array}\]

Therefore, 176 is the sum.

Practice Questions

1. The AP series is 8, 3, -2,.... Find the sum of the first 22 terms.

1. 879

2. 979

3. -979

4. 980

2. If the first term is 10 and the sum of the first 14 terms of an AP is 1050, find the 20th term.

1. 200

2. 215

3. 195

4. 210

Answers:

1. Option (c)

2. Option (a)

Conclusion

As discussed above, an arithmetic progression is a sequence of numbers in order, in which the difference between any two consecutive numbers is a constant value. This article provides you with the simple formula used for finding the sum of the first n terms of an AP, simple steps to determine the sum, the formula for the sum of n natural numbers, the formula for the sum of the square of n term, and shortcut tricks for finding the sum. Some solved examples are provided in this article to boost your concept and clear your doubt about the topic.