<p>Solve the equation \((x+1)+(x+4)+(x+7)+\cdots+(x+28)=155\).</p>
Step-by-Step Solution
Key Concept: Recognize this as an arithmetic sequence with first term (x+1), last term (x+28), and common difference 3. Use the arithmetic series sum formula: S_n = (n/2)(first + last term).
<p><strong>Step 1:</strong> Identify the arithmetic sequence. First term = (x+1), common difference d = 3, last term = (x+28).</p><p><strong>Step 2:</strong> Find the number of terms: The general term is (x+1) + (n-1)·3 = x+28, so 3(n-1) = 27, giving n = 10 terms.</p><p><strong>Step 3:</strong> Apply the arithmetic series sum formula: S_n = (n/2)(first term + last term)</p><p>155 = (10/2)[(x+1) + (x+28)]</p><p>155 = 5(2x + 29)</p><p>155 = 10x + 145</p><p><strong>Step 4:</strong> Solve for x: 10x = 10, so x = 1</p><p><strong>Verification:</strong> Sum = 2 + 5 + 8 + 11 + 14 + 17 + 20 + 23 + 26 + 29 = 155 ✓</p><p>∴ Answer: x = 1</p>
Correct Answer: x = 1