<p>Find the <em>n</em>th term and the sum to <em>n</em> terms of the series: <br> \(2 + 5 + 12 + 31 + 86 + \ldots\)</p>
Step-by-Step Solution
Key Concept: Decompose the series into two parts by recognizing the differences form a pattern: each term equals n plus a power of 3, i.e., aₙ = n + 3^(n-1). Then sum the arithmetic and geometric parts separately.
<p><strong>Step 1: Find the pattern by examining differences</strong></p><p>Given series: 2, 5, 12, 31, 86, ...</p><p>First differences: 3, 7, 19, 55, ... = 3¹, 3¹+4, 3²+10, 3³+28, ...</p><p>Notice: 2 = 1 + 3⁰, 5 = 2 + 3¹, 12 = 3 + 3², 31 = 4 + 3³, 86 = 5 + 3⁴</p><p><strong>Step 2: Identify the general term</strong></p><p>aₙ = n + 3^(n-1)</p><p><strong>Step 3: Find sum to n terms by splitting</strong></p><p>Sₙ = Σ(k=1 to n) aₖ = Σ(k=1 to n) k + Σ(k=1 to n) 3^(k-1)</p><p>First sum (AP): Σk = n(n+1)/2</p><p>Second sum (GP with a=1, r=3): Σ3^(k-1) = (3ⁿ - 1)/(3 - 1) = (3ⁿ - 1)/2</p><p><strong>Step 4: Combine results</strong></p><p>Sₙ = n(n+1)/2 + (3ⁿ - 1)/2 = (1/2){n(n+1) + 3ⁿ - 1}</p><p>∴ <strong>nth term = n + 3^(n-1)</strong></p><p>∴ <strong>Sum = (1/2){n(n+1) + 3ⁿ - 1}</strong></p>
Correct Answer: nth term = n + 3^{n-1}, Sum = (1/2){n(n+1) + 3^n - 1}