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Binomial Theorem · Paper 3 — Free Maths AI HL Practice Questions

1ChallengeLAQBinomial expansion and coefficients10 marksPaper 3~15 min
A mathematics student investigates the Catalan numbers, defined for n∈N, n≥2n \in \mathbb{N},\ n \geq 2 by: an=1n+1(2nn)a_n = \frac{1}{n+1}\binom{2n}{n}
(a)
Show that an=(2nn)−(2nn+1)a_n = \dbinom{2n}{n} - \dbinom{2n}{n+1}. [4 marks]
(b)
Show that an+1an=2(2n+1)(n+1)(n+2)⋅(n+1)=4n+2n+2\dfrac{a_{n+1}}{a_n} = \dfrac{2(2n+1)}{(n+1)(n+2)} \cdot (n+1) = \dfrac{4n+2}{n+2}. [3 marks]
(c)
The student claims that {an}\{a_n\} is strictly decreasing for n≥2n \geq 2. Determine whether this claim is correct. Hence, by considering an+1an\dfrac{a_{n+1}}{a_n} as n→∞n \to \infty, describe the long-run behaviour of consecutive terms and explain what this implies about the rate of growth of ana_n. [3 marks]

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