Arithmetic Progressions

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An Arithmetic Progression (AP) is a sequence of numbers where each term after the first is obtained by adding a fixed number to the preceding term. This fixed number is called the common difference. E

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The common difference (d) is the fixed number that is added to each term to get the next term. It can be positive, negative, or zero. Formula: d = a₂ - a₁ = a₃ - a₂ = ... = aₙ₊₁ - aₙ

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The general form of an AP is: a, a+d, a+2d, a+3d, ... where 'a' is the first term and 'd' is the common difference.

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Common difference (d) = 7 - 3 = 4. We can verify: 11 - 7 = 4 and 15 - 11 = 4. Since the difference is constant, d = 4.

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The nth term of an AP is given by: aₙ = a + (n-1)d where a = first term, d = common difference, n = position of the term. This is also called the general term.

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Given: a = 5, d = 8 - 5 = 3, n = 10. Using aₙ = a + (n-1)d: a₁₀ = 5 + (10-1)×3 = 5 + 9×3 = 5 + 27 = 32. Therefore, the 10th term is 32.

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To check if a sequence is an AP, find the differences between consecutive terms. If all differences are equal, it's an AP. Example: For 2, 5, 8, 11: 5-2=3, 8-5=3, 11-8=3. Since all differences equal 3

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Finite AP: Has a limited number of terms with a last term. Example: 3, 6, 9, 12, 15 (5 terms). Infinite AP: Continues indefinitely without a last term. Example: 1, 2, 3, 4, ... (continues forever).