1
IIT-JEE 1989
Subjective
+5
-0
Prove that
$${C_0} - {2^2}{C_1} + {3^2}{C_2}\,\, - \,..... + {\left( { - 1} \right)^n}{\left( {n + 1} \right)^2}{C_n} = 0,\,\,\,\,n > 2,\,\,$$ where $${C_r} = {}^n{C_r}.$$
2
IIT-JEE 1988
Subjective
+5
-0
Let $$R$$ $$ = {\left( {5\sqrt 5 + 11} \right)^{2n + 1}}$$ and $$f = R - \left[ R \right],$$ where [ ] denotes the greatest integer function. Prove that $$Rf = {4^{2n + 4}}$$
3
IIT-JEE 1987
Subjective
+3
-0
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 $$n$$.
4
IIT-JEE 1985
Subjective
+5
-0
Use method of mathematical induction $${2.7^n} + {3.5^n} - 5$$ is divisible by $$24$$ for all $$n > 0$$
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