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

1ChallengeLAQBinomial expansion and coefficientss10 marksPaper 3~15 min

The binomial theorem states that for any positive integer

n

(1+x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k.

Consider the sum

S_n

defined for integers

n \geq 1

S_n = \sum_{k=0}^{n} \frac{(-1)^k}{k+1}\binom{n}{k}.

(a)

Prove that

S_n = \dfrac{1}{n+1}

for all integers

n \geq 1

. [5 marks]

(b)

Let

T_n = \displaystyle\sum_{k=0}^{n} \dfrac{(-1)^k}{k+2}\binom{n}{k}

. By writing

\dfrac{1}{k+2}\binom{n}{k}

in terms of

\binom{n+1}{k+1}

or otherwise, express

T_n

in terms of

S_{n+1}

and hence find a closed form for

T_n

in terms of

n

. Justify each step of your reasoning. [5 marks]

Solutions

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