<p>The ratio of the sums of \(n\) terms of two APs is \((3n-13):(5n+21)\). Find the ratio of the 24th terms of the two progressions.</p>
Step-by-Step Solution
Key Concept: The sum of n terms of an AP can be expressed as S_n = (n/2)(2a + (n-1)d) = (n/2)(first term + last term). Use the property that the mth term equals (S_{2m-1})/(2m-1) to convert the sum ratio into a term ratio.
<p><strong>Step 1:</strong> For two APs with sums S₁ₙ and S₂ₙ, we're given:</p><p>S₁ₙ : S₂ₙ = (3n - 13) : (5n + 21)</p><p><strong>Step 2:</strong> Use the key property: For an AP, the mth term a_m = S_{2m-1}/(2m-1)</p><p>This is because S_{2m-1} = [(2m-1)/2](first term + last term) = [(2m-1)/2](a₁ + a_{2m-1}), and the middle term of 2m-1 consecutive terms is a_m.</p><p><strong>Step 3:</strong> For the 24th term, substitute n = 2(24) - 1 = 47:</p><p>a₁₂₄ : a₂₂₄ = S₁₄₇ : S₂₄₇ = [3(47) - 13] : [5(47) + 21]</p><p><strong>Step 4:</strong> Calculate:</p><p>= (141 - 13) : (235 + 21)</p><p>= 128 : 256</p><p>= 1 : 2</p><p><strong>∴ Answer: 1 : 2</strong></p>
Correct Answer: 1