Progressions

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An Arithmetic Progression is a sequence of numbers in which each term (except the first) is obtained by adding a fixed number to the preceding term. This fixed number is called the common difference.

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The common difference (d) is found by subtracting any term from the next term: d = a₂ - a₁ = a₃ - a₂ = aₖ₊₁ - aₖ Example: In A.P. 2, 5, 8, 11, ... d = 5 - 2 = 3 or d = 8 - 5 = 3 Note: The common dif

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Formula: aₙ = a + (n - 1)d Where: • aₙ = nth term • a = first term • n = position of the term • d = common difference This formula allows us to find any term in an A.P. without listing all previous

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Given: a = 3, d = 7 - 3 = 4, n = 20 Using aₙ = a + (n - 1)d: a₂₀ = 3 + (20 - 1) × 4 a₂₀ = 3 + 19 × 4 a₂₀ = 3 + 76 a₂₀ = 79 Therefore, the 20th term is 79.

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There are two formulas for the sum of first n terms: 1) Sₙ = n/2[2a + (n - 1)d] (when first term and common difference are known) 2) Sₙ = n/2(a + l) (when first term and last term are known)

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Given: a = 5, d = 8 - 5 = 3, n = 20 Using Sₙ = n/2[2a + (n - 1)d]: S₂₀ = 20/2[2(5) + (20 - 1) × 3] S₂₀ = 10[10 + 19 × 3] S₂₀ = 10[10 + 57] S₂₀ = 10 × 67 S₂₀ = 670 Therefore, the sum of first 20 term

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A Geometric Progression is a sequence of non-zero numbers in which each term (except the first) is obtained by multiplying the preceding term by a fixed non-zero number. This fixed number is called th

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The common ratio (r) is found by dividing any term by the preceding term: r = a₂/a₁ = a₃/a₂ = aₖ₊₁/aₖ Example: In G.P. 2, 6, 18, 54, ... r = 6/2 = 3 or r = 18/6 = 3 Note: The common ratio is constan