Mathematics Class 10 - Arithmetic Progressions Notes

Arithmetic Progressions

In this chapter, you will learn about arithmetic progressions (AP), their properties, how to find terms and sums, and how to solve problems using AP formulas. By the end of this chapter, you will be able to identify, analyze, and solve questions related to arithmetic progressions in real-life and mathematical situations.

Key Concepts

• Arithmetic Progression (AP): A sequence of numbers in which the difference between any two consecutive terms is constant.
• First Term (a): The first number in the sequence.
• Common Difference (d): The fixed difference between consecutive terms.
• n^th Term (a_n): The term at position n in the sequence.

General Form of an AP

An arithmetic progression looks like:
a, a + d, a + 2d, a + 3d, ..., a + (n-1)d

n^th Term of an AP

The n^th term of an AP is given by:
a_n = a + (n - 1)d

• a = first term
• d = common difference
• n = number of terms

Example: If a = 3, d = 5, n = 4,
a₄ = 3 + (4 - 1) × 5 = 3 + 15 = 18

Sum of First n Terms of an AP

The sum of the first n terms (S_n) is:
S_n = n/2 × [2a + (n - 1)d]
or
S_n = n/2 × (a + l), where l is the last term.

• a = first term
• d = common difference
• n = number of terms
• l = last term

Example: Find the sum of the first 5 terms of the AP: 2, 4, 6, 8, 10.
Here, a = 2, d = 2, n = 5
S₅ = 5/2 × [2 × 2 + (5 - 1) × 2] = 5/2 × [4 + 8] = 5/2 × 12 = 30

Properties and Applications

• If the common difference (d) is positive, the AP increases; if negative, it decreases.
• APs are used in real life for patterns, schedules, and planning (e.g., saving money, arranging seats, etc.).
• The difference between any two consecutive terms is always the same in an AP.

Practice Questions

1. Find the 10^th term of the AP: 7, 10, 13, 16, ...
2. What is the sum of the first 8 terms of the AP: 5, 9, 13, ...?
3. If the 5^th term of an AP is 20 and the common difference is 3, what is the first term?
4. Which term of the AP 4, 9, 14, 19, ... is 64?
5. If the sum of the first n terms of an AP is 105, the first term is 5, and the common difference is 2, find n.

Challenge Yourself

• If the 3^rd term of an AP is 12 and the 7^th term is 24, find the common difference and the first term.
• The sum of three numbers in AP is 27 and their product is 504. Find the numbers.

Did You Know?

• The sequence of even numbers (2, 4, 6, 8, ...) and odd numbers (1, 3, 5, 7, ...) are both APs.
• APs are used in computer science, finance, and many fields of science and engineering.

Glossary

• Arithmetic Progression (AP): A sequence with a constant difference between terms.
• Common Difference (d): The fixed amount added to each term to get the next.
• n^th Term: The term at position n in the sequence.
• Sum of AP: The total when all terms are added together.

Answers to Practice Questions

1. a = 7, d = 3; a₁₀ = 7 + (10-1)×3 = 7 + 27 = 34
2. a = 5, d = 4, n = 8; S₈ = 8/2 × [2×5 + (8-1)×4] = 4 × [10 + 28] = 4 × 38 = 152
3. a₅ = a + 4d = 20; d = 3 ⇒ a = 20 - 12 = 8
4. a = 4, d = 5; 64 = 4 + (n-1)×5 ⇒ (n-1)×5 = 60 ⇒ n-1 = 12 ⇒ n = 13
5. S_n = n/2 × [2×5 + (n-1)×2] = 105
   n/2 × [10 + 2n - 2] = 105
   n/2 × (2n + 8) = 105
   n(n + 4) = 105
   n² + 4n - 105 = 0
   (n + 15)(n - 7) = 0 ⇒ n = 7 (since n > 0)

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Practice solving AP problems to strengthen your understanding of sequences and series!