What is Arithmetic Mean?
The first two terms a and b will essentially have a difference, which will be equal to the next two terms b and c in the arithmetic progression if the three integers are in AP, which means if a, b, and c are in AP.
a − b = b − c
⇒ 2b = a + c
⇒ b = (a + b) / 2
Thus, the required arithmetic mean (A. M) of two number ‘a’ and ‘b’ is (a + b) / 2.
Some examples of arithmetic mean:
- The arithmetic mean between 12 and 32 in the arithmetic progression of 12, 22, and 32 is 22.
- The arithmetic mean between 7 and 11 in the arithmetic progression of 7, 9, and 11 is 9.
Example: In an A.M. the sum of three consecutive terms is −3 and their product is 8. Then find the terms.
Solution:
Let three consecutive terms are a, b and c.
It is given that the sum of three consecutive terms is −3.
So, a + b +c = −3.
It is given that the product of three consecutive terms is 8.
So, abc = 8.
According to the definition of A.M.,
2b = a + c
Substitute 2b for a + c in a + b +c = −3.
2b+ b = −3
⇒ 3b = −3
⇒ b = −1
Substitute −1 for b in 2b = a + c and then solve for a.
a + c = −2
⇒ a = −2 − c
Substitute −1 for b and −2 − c for a in abc = 8 and then solve for c.
(2 + c) c = 8
⇒ c2 + 2c − 8=0
⇒ c2 + 4c − 2c − 8 = 0
⇒ c(c + 4) − 2 (c + 4) = 0
⇒ (c + 4)(c − 2) = 0
⇒ c = − 4 or c = 2
Case 1: When c = − 4
Substitute −4 for c in a = −2 − c.
a = −2 + 4
⇒ a = 2
Case 2: When c = 2
Substitute 2 for c in a = −2 − c.
a = −2 − 2
⇒ a = −4
Therefore, the terms are −4, −1, 2 or 2, −1, −4.
Also, Read
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.