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 β b = b β c
β 2b = a + c
β b = (a + b) / 2
Thus, the required arithmetic mean (A. M) of two number βaβ and βbβ 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 β3 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 β3.
So, a + b +c = β3.
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 = β3.
2b+ b = β3
β 3b = β3
β b = β1
Substitute β1 for b in 2b = a + c and then solve for a.
a + c = β2
β a = β2 β c
Substitute β1 for b and β2 β c for a in abc = 8 and then solve for c.
(2 + c) c = 8
β c2 + 2c β 8=0
β c2 + 4c β 2c β 8 = 0
β c(c + 4) β 2 (c + 4) = 0
β (c + 4)(c β 2) = 0
β c = β 4 or c = 2
Case 1: When c = β 4
Substitute β4 for c in a = β2 β c.
a = β2 + 4
β a = 2
Case 2: When c = 2
Substitute 2 for c in a = β2 β c.
a = β2 β 2
β a = β4
Therefore, the terms are β4, β1, 2 or 2, β1, β4.
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 studentsβ 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.