Selection of terms in an A.P.
Number of terms
Terms
Common difference
3
a − d, a, a + d
d
4
a − 3d, a − d, a + d, a + 3d
2d
5
a − 2d, a − d, a, a + d, a + 2d
d
6
a − 5d, a − 3d, a − d , a + d, a + 3d, a + 5d
2d
It is noticed that if a number of terms is odd then the first term and common difference are a and d, if a number of terms are even then the first term and common difference are a and 2d.
Example: The sum of three numbers in A.P. is 12 and the sum of their cubes is 288. Find the numbers.
Solution:
Let a represents the first term and d represents the common difference.
So, the three numbers are a − d, a, and a + d.
It is given that the sum of three numbers is 12.
a − d + a + a + d = 12
⇒ 3a = 12
⇒ a = 4
It is given that the sum of their cubes is 288.
(a − d)3 + a3 + (a + d)3 = 288
Substitute 4 for a in the above equation.
(4 − d)3 + 43 + (4 + d)3 = 288
⇒ 43 − d3 − 3(4)(d)(4 − d) + 64 + 43 + d3 + 3(4)(d)(4 + d) = 288
⇒ 64 − d3 − 48d + 12d2 + 64 + 64 + d3 + 48d + 12d2 = 288
⇒ 24d2 = 96
⇒ d2 = 4
⇒ d = ±2
Case 1: When a = 4 and d = 2
The three numbers are 2, 4, and 6.
Case 2: When a = 4 and d = −2
The three numbers are 6, 4, and 2.
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.
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