<p>For an AP of odd number of terms, the sum of all the terms is \(\dfrac{15}{8}\) times the sum of the terms in odd places. Find the number of terms in the AP.</p>
Step-by-Step Solution
Key Concept: In an AP with odd number of terms (2n+1), the sum of terms in odd places forms an AP with (n+1) terms. Use the ratio of total sum to odd-place sum to establish an equation relating n to the given ratio 15/8.
<p><strong>Step 1:</strong> Let the AP have (2n+1) terms with first term 'a' and common difference 'd'.</p><p><strong>Step 2:</strong> Sum of all terms: S = (2n+1)/2 × [2a + 2nd] = (2n+1)(a + nd)</p><p><strong>Step 3:</strong> Terms in odd places are: a, a+2d, a+4d, ..., a+2nd. These form an AP with (n+1) terms.</p><p><strong>Step 4:</strong> Sum of odd-place terms: S_odd = (n+1)/2 × [2a + 2nd] = (n+1)(a + nd)</p><p><strong>Step 5:</strong> Given condition: S/S_odd = 15/8</p><p>[(2n+1)(a + nd)] / [(n+1)(a + nd)] = 15/8</p><p><strong>Step 6:</strong> Simplifying: (2n+1)/(n+1) = 15/8</p><p><strong>Step 7:</strong> Cross multiply: 8(2n+1) = 15(n+1)</p><p>16n + 8 = 15n + 15</p><p>n = 7</p><p><strong>Step 8:</strong> Number of terms = 2n + 1 = 2(7) + 1 = 15</p><p>∴ Answer: <strong>15</strong></p>
Correct Answer: 15