<p>If the sum of \(m\) terms of an A.P. is the same as the sum of its \(n\) terms, then the sum of its \((m+n)\) terms is</p>
Step-by-Step Solution
Key Concept: If S_m = S_n for an A.P., then S_m - S_n = 0, which means the sum of terms from position (m+1) to n equals zero. Use this constraint along with the general sum formula to find S_(m+n).
<p><strong>Step 1:</strong> Use the A.P. sum formula: S_k = (k/2)[2a + (k-1)d]</p><p><strong>Step 2:</strong> Given S_m = S_n, write:<br>S_m = (m/2)[2a + (m-1)d] = (n/2)[2a + (n-1)d] = S_n</p><p><strong>Step 3:</strong> Simplify: m[2a + (m-1)d] = n[2a + (n-1)d]<br>2am + m(m-1)d = 2an + n(n-1)d<br>2a(m-n) = n(n-1)d - m(m-1)d<br>2a(m-n) = d[n(n-1) - m(m-1)]<br>2a(m-n) = d[(n² - m²) - (n - m)]<br>2a(m-n) = d(m-n)(n+m-1)×(-1)</p><p><strong>Step 4:</strong> Since m ≠ n: 2a = -d(m+n-1) or 2a + d(m+n-1) = 0</p><p><strong>Step 5:</strong> Calculate S_(m+n):<br>S_(m+n) = ((m+n)/2)[2a + (m+n-1)d]<br>= ((m+n)/2)[2a + d(m+n-1)]<br>= ((m+n)/2) × 0<br>= 0</p><p><strong>∴ Answer: S_(m+n) = 0 (Option D)</strong></p>
Correct Answer: D