Sequences and Series

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A sequence is an ordered collection of numbers arranged according to a specific pattern or rule. Each number in the sequence is called a term. Example: 2, 4, 6, 8, 10, ... (even numbers) where a₁ = 2,…

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A sequence is an ordered list of terms (e.g., 1, 3, 5, 7, ...), while a series is the sum of terms in a sequence (e.g., 1 + 3 + 5 + 7 + ...). If a₁, a₂, a₃, ... is a sequence, then a₁ + a₂ + a₃ + ...

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An Arithmetic Progression is a sequence where each term (except the first) is obtained by adding a constant 'd' to the previous term. General form: a, a+d, a+2d, a+3d, ... where 'a' is the first term …

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The nth term of an AP is: aₙ = a + (n-1)d, where 'a' is the first term, 'd' is the common difference, and 'n' is the term number. This formula helps find any term in the AP without calculating all pre…

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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' is the first term, 'd' is the common difference, 'l' is the last term, and 'n' is the number of terms.

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Given: a = 3, d = 7-3 = 4, n = 15 Using aₙ = a + (n-1)d a₁₅ = 3 + (15-1)×4 = 3 + 14×4 = 3 + 56 = 59 Therefore, the 15th term is 59.

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Arithmetic Mean is the average of numbers. For two numbers a and b, AM = (a+b)/2. If A is the AM of a and b, then a, A, b forms an AP. The AM is the middle term that makes the sequence arithmetic.

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1. Adding a constant to each term: Results in AP with same common difference 2. Multiplying each term by a constant k: Results in AP with common difference = k×d 3. Equidistant terms property: In fini…