What is 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 nth 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 an=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.
879
979
-979
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.
200
215
195
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.
FAQs on Sum of N Terms
1. What is n in AP?
The full form of AP is Arithmetic Progression and n stands for the number of terms.
2. How do you find the sum of AP?
You can find the sum of AP by using the formula 2a+(n-1) d.
3. Derive the Sum of the First n Terms of an AP.
Let us take the first term = a
The second term = a+d and the Third term = a+2d
So, the nth term will be = a+(n-1) d
How do we find the sum of all terms?
In the terms of average,
We know that Average = (Sum of all the terms)/n
Therefore, Sum of AP series can be written as,
Sum of terms = n × Average --------> Equation 1
NOTE: The average of the evenly spaced numbers can be written as,
Average = (First term + Second term) / 2
Now, substituting the value of average in Equation 1 we get,
Sum of terms = n× (First term + Second term) / 2
By taking (n/2) common and replacing the last term, we get
Sum of n terms of AP = 2a+(n−1)d2a+(n−1)d
This is the formula that denotes the sum of a series.
4. What is the significance of Arithmetic Progression (AP) in the real-world?
Real-world applications of arithmetic progression include the analysis of trends like straight-line depreciation. It is sometimes used to make predictions, such as when a person is waiting for a cab and traffic is going steadily forward. The time when the following taxi will arrive can be determined. In designs like pyramids where the objects are constantly changing, metric progression is also used. It is also used in a variety of fields, including history, economics, and sociology.
5. What makes AP and GP different from one another?
In an arithmetic progression (AP), all of the differences between any two successive terms are the same, whereas in a geometric progression, all of the ratios between any two consecutive terms are the same (GP).