Arithmetic Progressions Solved Examples (Class 10 Maths)
An arithmetic progression is a sequence where consecutive terms have a constant difference. Learning to find the nth term, sum of terms, and solve problems
TL;DR: An arithmetic progression is a sequence where consecutive terms have a constant difference. Learning to find the nth term, sum of terms, and solve pro…
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An arithmetic progression is a sequence where consecutive terms have a constant difference. Learning to find the nth term, sum of terms, and solve problems
Arithmetic Progressions — Solved Numerical Examples (Step by Step)
Example 1: Find the 10th term of the arithmetic progression 2, 5, 8, 11, ...
Solution: Given:
First term a = 2
Common difference d = 5 - 2 = 3
n = 10
Using the formula: an = a + (n - 1)d
a10 = 2 + (10 - 1) × 3
a10 = 2 + 9 × 3
a10 = 2 + 27
a10 = 29
Example 2: Find the sum of the first 20 terms of the arithmetic progression 3, 7, 11, 15, ...
Solution: Given:
First term a = 3
Common difference d = 7 - 3 = 4
n = 20
Using the formula: Sn = n/2 × (2a + (n - 1)d)
S20 = 20/2 × (2 × 3 + (20 - 1) × 4)
S20 = 10 × (6 + 19 × 4)
S20 = 10 × (6 + 76)
S20 = 10 × 82
S20 = 820
Example 3: Is 50 a term of the arithmetic progression 1, 4, 7, 10, ...? If yes, find which term it is.
Solution: Given:
First term a = 1
Common difference d = 4 - 1 = 3
We need to check if 50 is a term.
Using: an = a + (n - 1)d
50 = 1 + (n - 1) × 3
50 = 1 + 3n - 3
50 = 3n - 2
52 = 3n
n = 52/3 = 17.33...
Since n is not a whole number, 50 is not a term of this progression.
Example 4: An arithmetic progression has first term 5 and last term 45. If there are 9 terms in total, find the sum.
Solution: Given:
First term a = 5
Last term l = 45
Number of terms n = 9
Using the formula: Sn = n/2 × (a + l)
S9 = 9/2 × (5 + 45)
S9 = 9/2 × 50
S9 = 9 × 25
S9 = 225
Example 5: Find the common difference of the arithmetic progression if the 5th term is 20 and the 10th term is 35.
Solution: Given:
a5 = 20
a10 = 35
Using the formula: an = a + (n - 1)d
a5 = a + 4d = 20 ... (1)
a10 = a + 9d = 35 ... (2)
Subtracting (1) from (2):
(a + 9d) - (a + 4d) = 35 - 20
5d = 15
d = 3
Example 6: The sum of the first n terms of an arithmetic progression is Sn = 3n^2 + 2n. Find the first three terms.
Solution: Given: Sn = 3n^2 + 2n
For n = 1: S1 = 3(1)^2 + 2(1) = 3 + 2 = 5
So a1 = 5
For n = 2: S2 = 3(2)^2 + 2(2) = 12 + 4 = 16
a2 = S2 - S1 = 16 - 5 = 11
For n = 3: S3 = 3(3)^2 + 2(3) = 27 + 6 = 33
a3 = S3 - S2 = 33 - 16 = 17
The first three terms are 5, 11, 17
Example 7: How many terms are there in the arithmetic progression 10, 13, 16, ..., 58?
Solution: Given:
First term a = 10
Common difference d = 13 - 10 = 3
Last term an = 58
Using: an = a + (n - 1)d
58 = 10 + (n - 1) × 3
58 = 10 + 3n - 3
58 = 7 + 3n
51 = 3n
n = 17
Tips
- Always identify the first term a and common difference d before using any formula. The common difference is found by subtracting any term from the next term.
- Use Sn = n/2 × (a + l) when you know the last term, as it is simpler than the other sum formula. Otherwise use Sn = n/2 × (2a + (n-1)d).
- To verify if a number is a term of an AP, substitute it in the formula an = a + (n-1)d and check if n comes out as a positive integer.
Frequently Asked Questions
What is the difference between the nth term and the sum of n terms?
The nth term (an) is just the value of that single term in the sequence. The sum of n terms (Sn) is the addition of all terms from the first to the nth term. For example, in 2, 5, 8, ..., the 3rd term is 8, but the sum of first 3 terms is 2 + 5 + 8 = 15.
How do I find the first term if only the sum formula is given?
If Sn is given, find S1 first because S1 = a1 (the first term). Then for the second term, use a2 = S2 - S1, and continue this pattern for other terms.
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