Sₙ ఇచ్చి a, d, n కనుగొనడం (Finding unknowns from the sum)

Correct: B — 200

1050 = 14/2 [20 + 13d] = 140 + 91d → 910 = 91d → d = 10. a₂₀ = 10 + 19×10 = 200.

Correct: C — 12

n/2 [18 + 8(n − 1)] = n(4n + 5) = 636 → 4n² + 5n − 636 = 0 → n = 12. (S₁₂ = 6 × (18 + 88) = 636.)

Correct: D — 8/3

S = n/2 (a + l) → 400 = n/2 × 50 → n = 16. l = a + 15d → 45 = 5 + 15d → d = 40/15 = 8/3.

Correct: A — 100

a + 3d = 7 (S₇ = 7(a + 3d)), a + 8d = 17 (S₁₇ = 17(a + 8d)) → d = 2, a = 1. Sₙ = n/2 [2 + 2(n−1)] = n² → S₁₀ = 100.

Correct: A — 4 లేదా 13

78 = n/2 (51 − 3n) → n² − 17n + 52 = 0 → (n − 4)(n − 13) = 0. రెండూ సరైనవి: 5వ నుండి 13వ పదం వరకు మొత్తం 0 కాబట్టి S₄ = S₁₃ = 78.

Correct: B — 8

a + 2d = 15; 10/2 (2a + 9d) = 125 → 2a + 9d = 25. 2a + 4d = 30 తో తీసివేయగా 5d = −5 → d = −1, a = 17. a₁₀ = 17 + 9(−1) = 8.

Correct: C — 85/3

9/2 (2a + 8×5) = 75 → 2a + 40 = 50/3 → a = −35/3. a₉ = a + 8d = −35/3 + 40 = 85/3.

Correct: D — 5

n/2 [4 + 8(n − 1)] = n(4n − 2) = 90 → 2n² − n − 45 = 0 → (2n + 9)(n − 5) = 0 → n = 5 (ఋణ/భిన్న విలువ −9/2 తిరస్కరణ).

Correct: A — 273

S₁₃ = 13/2 (a + a₁₃) = 13/2 × 42 = 13 × 21 = 273. (d = 28/12 = 7/3 అవసరం లేదు.)