Arithmetic Progressions

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An Arithmetic Progression is a sequence of numbers in which each term after the first is obtained by adding a fixed number (called common difference) to the preceding term. Example: 2, 5, 8, 11, 14...

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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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The nth term of an AP is given by: aₙ = a + (n-1)d where a = first term, d = common difference, n = term number. 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, then it's an AP. Example: For 3, 7, 11, 15... → 7-3=4, 11-7=4, 15-11=4. Since all differe…

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The sum of first n terms of an AP is: Sₙ = n/2[2a + (n-1)d] OR Sₙ = n/2(a + l) where a = first term, d = common difference, l = last term, n = number of terms…

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Use Sₙ = n/2(a + l) when you know the first term (a), last term (l), and number of terms (n), but the common difference is not given or not needed for calculation.