<p>In a non constant arithmetic progression having odd number of terms, having positive integral common difference, the ratio of the sum of the 1<sup>st</sup>, 3<sup>rd</sup>, 5<sup>th</sup>, 7<sup>th</sup>, ... terms to the sum of remaining terms is 13 : 12, then the number of terms in the arithmetic progression, is:</p>
Step-by-Step Solution
Key Concept: In an AP with odd number of terms, partition terms into odd-positioned (1st, 3rd, 5th, ...) and even-positioned terms. Use the sum formula for arithmetic progressions and set up a ratio equation to find the total number of terms.
<p><strong>Step 1: Set up the problem</strong><br/>Let the AP have n terms (n is odd), first term = a, common difference = d (positive integer).<br/>Terms: a, a+d, a+2d, ..., a+(n-1)d</p><p><strong>Step 2: Identify odd-positioned and even-positioned terms</strong><br/>Odd-positioned terms (1st, 3rd, 5th, ...): These are at positions 1, 3, 5, ..., n<br/>Number of odd-positioned terms = (n+1)/2<br/>Even-positioned terms (2nd, 4th, 6th, ...): These are at positions 2, 4, 6, ..., n-1<br/>Number of even-positioned terms = (n-1)/2</p><p><strong>Step 3: Find sum of odd-positioned terms</strong><br/>Odd-positioned terms form an AP: a, a+2d, a+4d, ..., a+(n-1)d<br/>This is an AP with first term = a, common difference = 2d, and (n+1)/2 terms<br/>S₁ = [(n+1)/2][2a + ((n+1)/2 - 1)·2d] = [(n+1)/2][2a + (n-1)d]</p><p><strong>Step 4: Find sum of even-positioned terms</strong><br/>Even-positioned terms form an AP: a+d, a+3d, a+5d, ..., a+(n-2)d<br/>This is an AP with first term = a+d, common difference = 2d, and (n-1)/2 terms<br/>S₂ = [(n-1)/2][2(a+d) + ((n-1)/2 - 1)·2d] = [(n-1)/2][2a + 2d + (n-3)d] = [(n-1)/2][2a + (n-1)d]</p><p><strong>Step 5: Set up the ratio equation</strong><br/>Given: S₁/S₂ = 13/12<br/>[(n+1)/2][2a + (n-1)d] / {[(n-1)/2][2a + (n-1)d]} = 13/12<br/>(n+1)/(n-1) = 13/12</p><p><strong>Step 6: Solve for n</strong><br/>12(n+1) = 13(n-1)<br/>12n + 12 = 13n - 13<br/>12 + 13 = 13n - 12n<br/>25 = n</p><p><strong>Step 7: Verify</strong><br/>With n = 25 (odd, ✓), odd-positioned terms = 13, even-positioned terms = 12<br/>Ratio of counts = 13:12 ✓<br/>Both counts match the given ratio, confirming our answer.</p><p><strong>∴ Answer: C</strong></p>
Correct Answer: C