Sequences and Series

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A sequence is a set of numbers arranged in a definite order, written as t₁, t₂, t₃, ..., tₙ. A series is the sum of terms of a sequence. For example: Sequence: 2, 4, 6, 8, ... Series: 2 + 4 + 6 + 8 + …

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An A.P. is a sequence where the difference between consecutive terms is constant. General form: a, a+d, a+2d, a+3d, ... where 'a' is first term and 'd' is common difference. Example: 3, 7, 11, 15, ...

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Step 1: Identify a = 5, d = 12 - 5 = 7 Step 2: Use formula tₙ = a + (n-1)d Step 3: t₁₅ = 5 + (15-1)×7 = 5 + 14×7 = 5 + 98 = 103 Therefore, the 15th term is 103.

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Formula: Sₙ = n/2[2a + (n-1)d] or Sₙ = n/2[first term + last term] Derivation: Write Sₙ = a + (a+d) + ... + (a+(n-1)d) Reverse: Sₙ = (a+(n-1)d) + ... + (a+d) + a Adding: 2Sₙ = n[2a + (n-1)d] Therefore…

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A G.P. is a sequence where the ratio of consecutive terms is constant. General form: a, ar, ar², ar³, ... where 'a' is first term and 'r' is common ratio (r ≠ 0). Example: 3, 6, 12, 24, ... (here a=3,…

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Step 1: Find a = 2, r = -6/2 = -3 Step 2: t₈ = ar⁷ = 2×(-3)⁷ = 2×(-2187) = -4374 Step 3: S₈ = a(r⁸-1)/(r-1) = 2((-3)⁸-1)/(-3-1) = 2(6561-1)/(-4) = 2×6560/(-4) = -3280 Therefore: t₈ = -4374 and S₈ = -3…

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Sum to infinity exists when |r| < 1 (common ratio's absolute value is less than 1). Formula: S∞ = a/(1-r) Example: For G.P. 1, 1/2, 1/4, 1/8, ... (a=1, r=1/2) S∞ = 1/(1-1/2) = 1/(1/2) = 2…

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Step 1: 0.36̄ = 0.36 + 0.0036 + 0.000036 + ... Step 2: This is G.P. with a = 0.36, r = 0.01 Step 3: Since |r| = 0.01 < 1, sum exists Step 4: S∞ = a/(1-r) = 0.36/(1-0.01) = 0.36/0.99 = 36/99 = 4/11 The…