Arithmetic and Geometric Progression

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A sequence is an arrangement of numbers according to a certain rule. For example: 2, 4, 6, 8, 10... (even numbers arranged in ascending order). Each number in a sequence is called a term.

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An Arithmetic Progression is a sequence where the difference between any two consecutive terms is constant. This constant difference is called the common difference (d). For 3, 7, 11, 15, 19: 7-3 = 4…

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The nth term of an A.P. is given by: Tₙ = a + (n-1)d Where: • a = first term • n = term number • d = common difference • Tₙ = nth term Example: For A.P. 5, 8, 11, 14... a = 5, d = 3 4th term: T₄ = 5…

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Given A.P.: 2, 5, 8, 11, ... First term (a) = 2 Common difference (d) = 5 - 2 = 3 Using formula: Tₙ = a + (n-1)d T₁₀ = 2 + (10-1) × 3 T₁₀ = 2 + 9 × 3 T₁₀ = 2 + 27 = 29 Therefore, the 10th term is 29…

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The sum of first n terms of an A.P. is: Sₙ = n/2 [2a + (n-1)d] Alternatively: Sₙ = n/2 (a + l) Where: • a = first term • d = common difference • l = last term • n = number of terms Use the first fo…

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Given A.P.: 4, 7, 10, 13, ... a = 4, d = 7 - 4 = 3, n = 15 Using formula: Sₙ = n/2 [2a + (n-1)d] S₁₅ = 15/2 [2(4) + (15-1) × 3] S₁₅ = 15/2 [8 + 14 × 3] S₁₅ = 15/2 [8 + 42] S₁₅ = 15/2 × 50 S₁₅ = 15 × …

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If three numbers a, b, c are in A.P., then: b - a = c - b This gives us: 2b = a + c The middle term is the arithmetic mean of the other two terms. Example: In A.P. 5, 8, 11: 2 × 8 = 16 and 5 + 11 =…

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The nth term from the end of an A.P. is: Tₙ (from end) = l - (n-1)d Where: • l = last term of the A.P. • d = common difference • n = position from the end Example: In A.P. 2, 5, 8, 11, 14, 17, 20 To…