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))

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

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

$$2028=4·507,so507/2028=1/4$$

$$507·S₂₀₂₅=2700/4=675$$

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