Question 12

For positive integers $$n$$, if $$4a_{n}=(n^{2}_5n+6)$$ and $$S_{n}= \sum_{k=1}^{n}\left(\frac{1}{a_{k}}\right)$$, then the value of $$507S_{2025}$$ is :

4aₙ = n²+5n+6 = (n+2)(n+3), so aₙ = (n+2)(n+3)/4

1/aₙ = 4/((n+2)(n+3)) = 4(1/(n+2) - 1/(n+3))

Sₙ = 4Σ(1/(k+2)-1/(k+3)) for k=1 to n = 4(1/3 - 1/(n+3))

S₂₀₂₅ = 4(1/3 - 1/2028) = 4·(2028-3)/(3·2028) = 4·2025/(3·2028) = 4·675/2028 = 2700/2028

507·S₂₀₂₅ = 507·2700/2028 = 507·2700/2028

2028 = 4·507, so 507/2028 = 1/4

507·S₂₀₂₅ = 2700/4 = 675

The correct answer is Option 2: 675.

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