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Home » \[ a_n = S_n - S_n-1 \] - RTA

\[ a_n = S_n - S_n-1 \] - RTA

📅 May 9, 2026 👤 scraface

May 09, 2026

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Substitute: \( 210 = \frac{n}{2}(2 \times 3 + (n-1) \times 2) = \frac{...
Simplify: \( 210 = n(n + 2) \) → \( n^2 + 2n - 210 = 0 \).
Solve quadratic: \( n = \frac{-2 \pm \sqrt{4 + 840}}{2} = \frac{-2 \pm...
Positive root: \( n = \frac{27.02}{2} \approx 13.51 \), so \( n = 14 \...
Verify: \( S_{14} = \frac{14}{2}(6 + 26) = 7 \times 32 = 224 \) → too...
Try \( n = 13 \): \( S_{13} = \frac{13}{2}(6 + 24) = 6.5 \times 30 = 1...
Recheck equation: \( n(n + 2) = 210 \) → \( n^2 + 2n - 210 = 0 \).
Discriminant: \( 4 + 840 = 844 \), \( \sqrt{844} \approx 29.05 \), so...
Wait: \( n(n+2) = 210 \), try factor pairs: \( 14 \times 15 = 210 \),...
\( 14 \times 16 = 224 \), \( 13 \times 15 = 195 \), no integer solutio...

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