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1

IIT-JEE 1989

Subjective
Using mathematical induction, prove that $${}^m{C_0}{}^n{C_k} + {}^m{C_1}{}^n{C_{k - 1}}\,\,\, + .....{}^m{C_k}{}^n{C_0} = {}^{\left( {m + n} \right)}{C_k},$$
where $$m,\,n,\,k$$ are positive integers, and $${}^p{C_q} = 0$$ for $$p < q.$$

Solve it.
2

IIT-JEE 1988

Subjective
Let $$R$$ $$= {\left( {5\sqrt 5 + 11} \right)^{2n + 1}}$$ and $$f = R - \left[ R \right],$$ where [ ] denotes the greatest integer function. Prove that $$Rf = {4^{2n + 4}}$$

Solve it.
3

IIT-JEE 1987

Subjective
Prove by mathematical induction that $$- 5 - {{\left( {2n} \right)!} \over {{2^{2n}}{{\left( {n!} \right)}^2}}} \le {1 \over {{{\left( {3n + 1} \right)}^{1/2}}}}$$ for all positive integers $$n$$.

Solve it.
4

IIT-JEE 1985

Subjective
Use method of mathematical induction $${2.7^n} + {3.5^n} - 5$$ is divisible by $$24$$ for all $$n > 0$$

Solve it.

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