Sum of n terms of an A.P.
The sum Sn of n terms of an A.P. with the first term a and common difference d is,
The sum Sn of n terms of an A.P. with last term l and common difference d is,
Note: The nth term of a of the sequence is calculated by an = Sn − Sn-1
Sum of First n Natural Numbers
Example: Find the sum of all 2-digit natural numbers divisible by 4.
Solution:
The smallest 2-digit natural number is 10.
The largest 2-digit natural number is 99.
The smallest 2-digit natural number which is divisible by 4 is 12.
So, a = 12
The largest 2-digit natural number which is divisible by 4 is 96.
So, an = 96
The sequence of 2-digit natural number which are divisible by 4 are 12, 16, 20, 24, … , 96.
Now, determine the common difference d.
d = 16 − 12
⇒ d = 4
Substitute 4 for d, 12 for a and 96 for an in an = a + (n − 1) d.
96 = 12 + (n − 1) 4
⇒ 96 = 12 + 4n -4
⇒ 4n = 96 −12 + 4
⇒ 4n = 88
⇒ n = 22
Substitute 4 for d, 12 for a and 22 for n in Sn = n / 2 (2a + (n − 1) d )
S22 = 22/2 (2(12)+(22 − 1) 4)
⇒ S22 = 11 (24 + (21) 4)
⇒ S22 = 11 (24 + 84)
⇒ S22 = 1188
Therefore, the sum of all 2-digit natural numbers divisible by 4 is 1188.
Arithmetic Progressions Class 10 Maths Notes Chapter 5
CBSE Class 10 Maths Notes Chapter 4 Arithmetic Progressions are an outstanding resource created by our team of knowledgeable Subject Experts at GfG. As ardent supporters of students’ education, we place a high priority on their learning and development, which is why we have written these in-depth notes to aid them in comprehending the challenging subject of arithmetic progressions.
Chapter 4 of the NCERT Class 10 Maths textbook finds the nth term of an arithmetic progression, summing the n terms of an arithmetic progression, calculating the arithmetic mean, and many other topics covered. These notes are intended to give students a thorough overview of the entire chapter, covering all the crucial topics, formulas, and ideas they will need to know to ace their examinations.