Arithmetic Progression

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A sequence is a set of numbers arranged in a definite order where each number has a specific position. Example: 1, 2, 3, 4, ... (natural numbers) or 1, 4, 9, 16, ... (perfect squares). In a sequence, …

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An Arithmetic Progression (A.P.) is a sequence where the difference between consecutive terms is constant. Condition: If t_{n+1} - t_n = d (constant) for all n, then the sequence is an A.P. The consta…

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Common difference (d) is the constant difference between consecutive terms in an A.P., calculated as d = t_{n+1} - t_n. Yes, it can be: Positive (increasing A.P.), Negative (decreasing A.P.), or Zero …

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The general form of an A.P. is: a, (a + d), (a + 2d), (a + 3d), ... Where: First term = a, Second term = a + d, Third term = a + 2d, Fourth term = a + 3d, and so on.

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The nth term of an A.P. is given by: t_n = a + (n - 1)d, where: a = first term, d = common difference, n = position of the term. This formula helps find any term in the A.P. without writing all previo…

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Given A.P.: 7, 10, 13, 16, ... First term a = 7, Common difference d = 10 - 7 = 3, n = 15. Using t_n = a + (n - 1)d: t₁₅ = 7 + (15 - 1) × 3 = 7 + 14 × 3 = 7 + 42 = 49. Therefore, the 15th term is 49.

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To check if a number k is in an A.P. with first term a and common difference d: 1. Use the formula k = a + (n - 1)d, 2. Solve for n: n = (k - a)/d + 1, 3. If n is a positive integer, then k is in the …

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Given A.P.: 5, 8, 11, 14, ... Here a = 5, d = 3. If 61 is the nth term: 61 = 5 + (n - 1) × 3, 61 = 5 + 3n - 3, 61 = 2 + 3n, 59 = 3n, n = 59/3 = 19.67... Since n is not a positive integer, 61 is NOT a …