Solved Examples on Sum of n Terms
Example 1: Consider the AP = 2, 4, 6, 8, 10,… Find the sum of the first 20 terms of this A.P.
Solution:
Given AP,
2, 4, 6, 8, 10,…
First Term (a) = 2
Common Difference (d) = 4 – 2 = 6 – 4 = 2
Sum of n terms is,
Sn = n/2 [2a + (n-1)d]
n = 20
Sn = 20/2 [2×2 + (20-1)×2]
Sn = 10(42) = 420
Thus, the sum of the of first 20 terms in the sequence are, 420
Example 2: Calculate the sum of the first 16 terms of the AP: S = 98 + 95 + 92 +…
Solution:
Given AP,
S = 98 + 95 + 92 +…
First Term (a) = 98
Common Difference (d) = 95 – 98 = 92 – 95 = -3
Sum of n terms is,
Sn = n/2 [2a + (n-1)d]
n = 16
Sn = 16/2 [2×98 + (16-1)×-3]
Sn = 8(151) = 1208
Thus, the sum of the of first 20 terms in the sequence are, 1208
Example 3: Calculate the sum of the first 24 terms of the AP: S = 8 + 13 + 18 +…
Solution:
Given AP,
S = 8 + 13 + 18 +…
First Term (a) = 8
Common Difference (d) = 13 – 8 = 18 – 13 = 5
Sum of n terms is,
Sn = n/2 [2a + (n-1)d]
n = 24
Sn = 24/2 [2×8 + (24-1)×5]
Sn = 8(131) = 1572
Thus, the sum of the of first 24 terms in the sequence are, 1572
Example 4: Find the sum of squares of the first 10 natural numbers.
Solution:
We know that formula for the sum of n terms of the square of the natural number is,
Sn = n(n+1)(2n+1)/6
For finding the square of the sum of the first 10 natural numbers
Sn = 10(10+1)(20+1)6
= 10(11)(21)/6
= 385
Thus, the sum of the first 10 squares of the natural number is 385.
Sum of N Terms of an AP
Arithmetic Sequence is defined as the sequence of numbers such that the difference between any two consecutive numbers is always constant. That is we can say that the next number in the arithmetic sequence is always the previous sequence plus the constant term, where the constant term can be both positive or negative. We can easily find the sum of the first “n” terms of the arithmetic sequence using the formula discussed in the article.
Johann Carl Friedrich Gauss in the 19th century was the first to find the sum of the first “n” terms of the arithmetic sequence where n is any natural number. Let us learn all about the sum of n terms of an AP along with examples in this article.
Table of Content
- What is Sum of n Terms of an AP?
- Sum of n Terms of AP Formula
- Sum of n Terms of AP Proof
- Sum of AP Formula for an Infinite AP
- Sum of AP Formulas
- Sum of Natural Numbers
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