Arithmetic Sequences and Algebra

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The algebraic form of any arithmetic sequence is xₙ = an + b, where: • xₙ is the nth term • n is the position number (1, 2, 3, ...) • a is the common difference • b is a constant Example: For sequenc…

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Step-by-step solution: Step 1: Find the common difference d = 9 - 5 = 4 Step 2: Use the general form xₙ = an + b Here, a = 4 (common difference) Step 3: Find b using the first term First term = 5 5 …

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Formula: nth term = f + (n - 1)d Step-by-step method: Step 1: Identify first term (f) and common difference (d) Step 2: Calculate position change from 1st to nth term = (n - 1) Step 3: Multiply: (n -…

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Step-by-step conversion: Starting formula: f + (n-1)d Step 1: Expand (n-1)d f + (n-1)d = f + nd - d Step 2: Rearrange terms = nd + (f - d) = dn + (f - d) Step 3: Compare with an + b • a = d (coeffi…

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Step-by-step solution: Given: x₁ = 5, x₁₀ = 23 Step 1: Find common difference Using xₙ = f + (n-1)d 23 = 5 + (10-1)d 23 = 5 + 9d 18 = 9d d = 2 Step 2: Write algebraic form Using xₙ = dn + (f-d) xₙ =…

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Formula: 1 + 2 + 3 + ... + n = ½n(n + 1) Derivation (Gauss method): Step 1: Write sum forward and backward S = 1 + 2 + 3 + ... + n S = n + (n-1) + (n-2) + ... + 1 Step 2: Add corresponding terms 2S …

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Step-by-step solution: Step 1: Find algebraic form First term = 4, common difference = 3 xₙ = 3n + 1 Step 2: Use sum formula for xₙ = an + b Sum = ½an(n+1) + bn Here a = 3, b = 1, n = 15 Step 3: Cal…

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Formula: Sum = ½an(n+1) + bn Derivation: Step 1: Write first n terms x₁ = a + b x₂ = 2a + b ... xₙ = na + b Step 2: Add all terms Sum = (a + 2a + ... + na) + (b + b + ... + b) = a(1 + 2 + ... + n) +…