<p>A G.P. consists of an even number of terms. If the sum of all the terms is 5 times the sum of the terms occupying odd places, then find the common ratio.</p>
Step-by-Step Solution
Key Concept: In a G.P. with even number of terms, separate the sum into odd-positioned and even-positioned terms, then use the given ratio condition to establish a relationship between the common ratio and the two sums.
<p><strong>Step 1:</strong> Let the G.P. have 2n terms with first term 'a' and common ratio 'r'.</p><p><strong>Step 2:</strong> Sum of all terms: S = a(r²ⁿ - 1)/(r - 1)</p><p><strong>Step 3:</strong> Terms at odd places (1st, 3rd, 5th, ..., (2n-1)th): a, ar², ar⁴, ..., ar^(2n-2). This is a G.P. with first term a, common ratio r², and n terms.</p><p>Sum of odd-placed terms: S₁ = a(r²ⁿ - 1)/(r² - 1)</p><p><strong>Step 4:</strong> Terms at even places (2nd, 4th, 6th, ..., 2nth): ar, ar³, ar⁵, ..., ar^(2n-1).</p><p>Sum of even-placed terms: S₂ = ar(r²ⁿ - 1)/(r² - 1)</p><p><strong>Step 5:</strong> Given condition: S = 5S₁</p><p>a(r²ⁿ - 1)/(r - 1) = 5 · a(r²ⁿ - 1)/(r² - 1)</p><p><strong>Step 6:</strong> Cancel a(r²ⁿ - 1):</p><p>1/(r - 1) = 5/(r² - 1) = 5/[(r - 1)(r + 1)]</p><p><strong>Step 7:</strong> Multiply both sides by (r - 1):</p><p>1 = 5/(r + 1)</p><p>r + 1 = 5</p><p>∴ Answer: r = 4</p>
Correct Answer: 4