isi-entrance 2015 Q1

isi-entrance · India · UGA 4 marks Sequences and Series Proof of Inequalities Involving Series or Sequence Terms

Define $a _ { n } = \left( 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } \right) ^ { n }$ and $b _ { n } = n ^ { n } ( n ! ) ^ { 2 }$. Recall $n !$ is the product of the first $n$ natural numbers. Then,
(a) $a _ { n } < b _ { n }$ for all $n > 1$
(b) $a _ { n } > b _ { n }$ for all $n > 1$
(c) $a _ { n } = b _ { n }$ for infinitely many $n$
(d) none of the above.

(b). Take the $n$th root of $a _ { n }$ and $b _ { n }$ and use A.M. $\geq$ G.M.

Define $a _ { n } = \left( 1 ^ { 2 } + 2 ^ { 2 } + \ldots + n ^ { 2 } \right) ^ { n }$ and $b _ { n } = n ^ { n } ( n ! ) ^ { 2 }$. Recall $n !$ is the product of the first $n$ natural numbers. Then,\\
(a) $a _ { n } < b _ { n }$ for all $n > 1$\\
(b) $a _ { n } > b _ { n }$ for all $n > 1$\\
(c) $a _ { n } = b _ { n }$ for infinitely many $n$\\
(d) none of the above.

Paper Questions

QB1 QB10 QB11 QB2 QB3 QB4 QB5 QB6 QB7 QB8 QB9 Q1 Q2 Q3 Q4 Q5 Q6 Q7 Q8 Q9 Q10 Q11 Q12 Q13 Q14 Q15 Q16 Q17 Q18 Q19 Q20 Q21 Q22 Q23 Q24 Q25 Q26 Q27 Q28