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📑 In This Chapter Guide (Table of Contents)
1. What is an Arithmetic Progression (AP)?
An Arithmetic Progression is a sequence of numbers in which each term is obtained by adding a fixed number d (called the Common Difference) to the preceding term, except the first term a.
a, a + d, a + 2d, a + 3d, ..., a + (n - 1)d
- Common difference
d = aₙ - aₙ₋₁can be positive (increasing AP), negative (decreasing AP), or zero (constant AP). - Three consecutive terms in AP are conveniently assumed as:
(a - d), a, (a + d). - Four consecutive terms in AP are assumed as:
(a - 3d), (a - d), (a + d), (a + 3d)(with common difference 2d).
2. General nth Term of an AP (aₙ)
The nth term of an AP with first term a and common difference d is given by:
aₙ = a + (n - 1) · d
nth Term from the END of an AP:
If an AP has last term l and common difference d, the nth term from the end is:
aₙ (from end) = l - (n - 1) · d
3. Sum of First n Terms of an AP (Sₙ)
The sum Sₙ of the first n terms of an AP is calculated using either of two formulas:
1. When common difference d is known: Sₙ = (n / 2) · [2a + (n - 1)d]
2. When first term a and last term l are known: Sₙ = (n / 2) · [a + l]
Crucial Relation Between aₙ and Sₙ:
The nth term is the difference between sum of n terms and sum of (n - 1) terms:
aₙ = Sₙ - Sₙ₋₁
Sum of First n Natural Numbers: Sₙ = 1 + 2 + 3 + ... + n = [n(n + 1)] / 2
💡 Frequently Asked Questions (FAQ)
❓ Which term of the AP 21, 18, 15, ... is -81?
Here a = 21, d = 18 - 21 = -3. Let an = -81. Using an = a + (n - 1)d: -81 = 21 + (n - 1)(-3) ⟹ -102 = -3(n - 1) ⟹ n - 1 = 34 ⟹ n = 35. The 35th term is -81.
❓ If the sum of first n terms of an AP is given by Sn = 3n² + 5n, find its 25th term.
Using an = Sn - Sn-1: an = (3n² + 5n) - [3(n-1)² + 5(n-1)] = 6n + 2. Substituting n = 25: a25 = 6(25) + 2 = 150 + 2 = 152.
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