Sequences and Series

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A sequence is an arrangement of numbers in a definite order according to some rule. It can be regarded as a function whose domain is the set of natural numbers or some subset of it. Example: 2, 4, 6, …

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A finite sequence contains a finite (limited) number of terms. Example: 2, 4, 8, 16, 32 (5 terms). An infinite sequence continues indefinitely without end. Example: 1, 3, 5, 7, 9, ... (all odd numbers…

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The Fibonacci sequence is: 1, 1, 2, 3, 5, 8, 13, ... It is generated by the recurrence relation: a₁ = a₂ = 1, and aₙ = aₙ₋₂ + aₙ₋₁ for n > 2. Each term is the sum of the two preceding terms.

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A series is the sum of terms of a sequence, expressed as a₁ + a₂ + a₃ + ... + aₙ + ... It can be finite or infinite depending on whether the corresponding sequence is finite or infinite. Series can be…

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A G.P. is a sequence where each term (except the first) bears a constant ratio to the preceding term. This constant is called the common ratio (r). General form: a, ar, ar², ar³, ..., where 'a' is the…

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The nth term of a G.P. is given by: aₙ = arⁿ⁻¹, where 'a' is the first term, 'r' is the common ratio, and 'n' is the term number. This formula allows us to find any term directly without calculating a…

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First term a = 3, common ratio r = 6/3 = 2. Using aₙ = arⁿ⁻¹: a₈ = 3 × 2⁸⁻¹ = 3 × 2⁷ = 3 × 128 = 384. Therefore, the 8th term is 384.

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For a G.P. with first term 'a' and common ratio 'r': If r = 1: Sₙ = na. If r ≠ 1: Sₙ = a(rⁿ - 1)/(r - 1) or Sₙ = a(1 - rⁿ)/(1 - r). The choice of formula depends on convenience of calculation.