<p>If the sum of first 10 terms of an A.P. is 4 times the sum of its first 5 terms, then find the ratio of first term and common difference.</p>
Step-by-Step Solution
Key Concept: Use the A.P. sum formula S_n = n/2[2a + (n-1)d] to create equations for S₁₀ and S₅, then exploit the given ratio condition S₁₀ = 4S₅ to find a relationship between a and d.
<p><strong>Step 1:</strong> Write the sum formula for A.P.</p><p>S_n = n/2[2a + (n-1)d]</p><p><strong>Step 2:</strong> Calculate S₁₀</p><p>S₁₀ = 10/2[2a + 9d] = 5(2a + 9d) = 10a + 45d</p><p><strong>Step 3:</strong> Calculate S₅</p><p>S₅ = 5/2[2a + 4d] = 5/2 · 2(a + 2d) = 5(a + 2d) = 5a + 10d</p><p><strong>Step 4:</strong> Apply the given condition S₁₀ = 4S₅</p><p>10a + 45d = 4(5a + 10d)</p><p>10a + 45d = 20a + 40d</p><p>45d - 40d = 20a - 10a</p><p>5d = 10a</p><p>d = 2a</p><p><strong>Step 5:</strong> Find the ratio a/d</p><p>a/d = a/(2a) = 1/2</p><p>∴ <strong>Answer: 1/2</strong></p>
Correct Answer: 1/2