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.

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.

The value of $\lim _ { n \rightarrow \infty } \frac { 1 ^ { 3 } + 2 ^ { 3 } + \ldots + n ^ { 3 } } { n ^ { 4 } }$ is:
(a) $\frac { 3 } { 4 }$
(b) $\frac { 1 } { 4 }$
(c) 1
(d) 4.

The value of $\lim _ { n \rightarrow \infty } \frac { 1 ^ { 3 } + 2 ^ { 3 } + \ldots + n ^ { 3 } } { n ^ { 4 } }$ is:
(a) $\frac { 3 } { 4 }$
(b) $\frac { 1 } { 4 }$
(c) 1
(d) 4.

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

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

For any $n \geq 5$, the value of $1 + \frac{1}{2} + \frac{1}{3} + \cdots + \frac{1}{2^n - 1}$ lies between
(A) 0 and $\frac{n}{2}$
(B) $\frac{n}{2}$ and $n$
(C) $n$ and $2n$
(D) none of the above

For any $n \geq 5$, the value of $1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \cdots + \frac { 1 } { 2 ^ { n } - 1 }$ lies between
(A) 0 and $\frac { n } { 2 }$
(B) $\frac { n } { 2 }$ and $n$
(C) $n$ and $2 n$
(D) none of the above

The value of $\lim_{n \rightarrow \infty} \frac{1^3 + 2^3 + \ldots + n^3}{n^4}$ is:
(A) $\frac{3}{4}$
(B) $\frac{1}{4}$
(C) 1
(D) 4

The value of $\lim _ { n \rightarrow \infty } \frac { 1 ^ { 3 } + 2 ^ { 3 } + \ldots + n ^ { 3 } } { n ^ { 4 } }$ is:
(A) $\frac { 3 } { 4 }$
(B) $\frac { 1 } { 4 }$
(C) 1
(D) 4

The integral part of $\sum _ { n = 2 } ^ { 9999 } \frac { 1 } { \sqrt { n } }$ equals
(A) 196
(B) 197
(C) 198
(D) 199 .

Let $a _ { n }$ be the number of subsets of $\{ 1,2 , \ldots , n \}$ that do not contain any two consecutive numbers. Then
(A) $a _ { n } = a _ { n - 1 } + a _ { n - 2 }$
(B) $a _ { n } = 2 a _ { n - 1 }$
(C) $a _ { n } = a _ { n - 1 } - a _ { n - 2 }$
(D) $a _ { n } = a _ { n - 1 } + 2 a _ { n - 2 }$.

Consider the sequence $1,2,2,3,3,3,4,4,4,4,5,5,5,5,5 , \ldots$ obtained by writing one 1 , two 2's, three 3's and so on. What is the $2020 ^ { \text {th} }$ term in the sequence?
(A) 62
(B) 63
(C) 64
(D) 65

The value of $$1 + \frac { 1 } { 1 + 2 } + \frac { 1 } { 1 + 2 + 3 } + \cdots + \frac { 1 } { 1 + 2 + 3 + \cdots 2021 }$$ is
(A) $\frac { 2021 } { 1010 }$.
(B) $\frac { 2021 } { 1011 }$.
(C) $\frac { 2021 } { 1012 }$.
(D) $\frac { 2021 } { 1013 }$.

Let us denote the fractional part of a real number $x$ by $\{ x \}$ (note: $\{ x \} = x - [ x ]$ where $[ x ]$ is the integer part of $x$ ). Then, $$\lim _ { n \rightarrow \infty } \left\{ ( 3 + 2 \sqrt { 2 } ) ^ { n } \right\}$$ (A) equals 0 .
(B) equals 1 .
(C) equals $\frac { 1 } { 2 }$.
(D) does not exist.

Consider a sequence $P_1, P_2, \ldots$ of points in the plane such that $P_1, P_2, P_3$ are non-collinear and for every $n \geq 4$, $P_n$ is the midpoint of the line segment joining $P_{n-2}$ and $P_{n-3}$. Let $L$ denote the line segment joining $P_1$ and $P_5$. Prove the following:
(a) The area of the triangle formed by the points $P_n, P_{n-1}, P_{n-2}$ converges to zero as $n$ goes to infinity.
(b) The point $P_9$ lies on $L$.

There is a rectangular plot of size $1 \times n$. This has to be covered by three types of tiles - red, blue and black. The red tiles are of size $1 \times 1$, the blue tiles are of size $1 \times 1$ and the black tiles are of size $1 \times 2$. Let $t _ { n }$ denote the number of ways this can be done. For example, clearly $t _ { 1 } = 2$ because we can have either a red or a blue tile. Also, $t _ { 2 } = 5$ since we could have tiled the plot as: two red tiles, two blue tiles, a red tile on the left and a blue tile on the right, a blue tile on the left and a red tile on the right, or a single black tile.
(a) Prove that $t _ { 2 n + 1 } = t _ { n } \left( t _ { n - 1 } + t _ { n + 1 } \right)$ for all $n > 1$.
(b) Prove that $t _ { n } = \sum _ { d \geq 0 } \binom { n - d } { d } 2 ^ { n - 2 d }$ for all $n > 0$.
Here,
$$\binom { m } { r } = \begin{cases} \frac { m ! } { r ! ( m - r ) ! } , & \text { if } 0 \leq r \leq m , \\ 0 , & \text { otherwise } , \end{cases}$$
for integers $m , r$.

Find, with proof, all possible values of $t$ such that
$$\lim _ { n \rightarrow \infty } \left\{ \frac { 1 + 2 ^ { 1/3 } + 3 ^ { 1/3 } + \cdots + n ^ { 1/3 } } { n ^ { t } } \right\} = c$$
for some real number $c > 0$. Also find the corresponding values of $c$.

Let $$S = \frac{1}{\sqrt{10000}} + \frac{1}{\sqrt{10001}} + \cdots + \frac{1}{\sqrt{160000}}$$ Then the largest positive integer not exceeding $S$ is
(A) 200
(B) 400
(C) 600
(D) 800

The limit $$\lim_{n \rightarrow \infty} \frac{2\log 2 + 3\log 3 + \cdots + n\log n}{n^2 \log n}$$ is equal to
(A) 0
(B) $1/4$
(C) $1/2$
(D) 1

For every increasing function $b : [1, \infty) \rightarrow [1, \infty)$ such that $$\int_1^\infty \frac{\mathrm{d}x}{b(x)} < \infty$$ we must have
(A) $\sum_{k=1}^{\infty} \frac{\sqrt{\log k}}{b(k)} < \infty$
(B) $\sum_{k=3}^{\infty} \frac{\log k}{b(\log k)} < \infty$
(C) $\sum_{k=1}^{\infty} \frac{e^k}{b\left(e^k\right)} < \infty$
(D) $\sum_{k=3}^{\infty} \frac{1}{\sqrt{b(\log k)}} < \infty$

Define $S _ { n } = \frac { 1 } { 2 } \cdot \frac { 3 } { 4 } \cdots \cdot \frac { 2 n - 1 } { 2 n }$ where $n$ is a positive integer. Then
(A) $S _ { n } < \frac { 1 } { \sqrt { 4 n + 2 } }$ for some $n > 2$.
(B) $S _ { n } < \frac { 1 } { \sqrt { 2 n + 1 } }$ for all $n \geq 2$.
(C) $S _ { n } < \frac { 1 } { \sqrt { 2 n + 5 } }$ for all $n \geq 2$.
(D) $S _ { n } > \frac { 1 } { \sqrt { 4 n + 2 } }$ for all $n \geq 2$.

For any $n \geq 5$, the value of $1 + \frac { 1 } { 2 } + \frac { 1 } { 3 } + \cdots + \frac { 1 } { 2 ^ { n } - 1 }$ lies between
(a) 0 and $n / 2$.
(B) $n / 2$ and $n$.
(C) $n$ and $2 n$.
(D) none of the above.

Let $$S _ { n } = \sum _ { k = 1 } ^ { n } \frac { n } { n ^ { 2 } + k n + k ^ { 2 } } \quad \text { and } \quad T _ { n } = \sum _ { k = 0 } ^ { n - 1 } \frac { n } { n ^ { 2 } + k n + k ^ { 2 } } ,$$ for $n = 1,2,3 , \cdots$. Then,
(A) $\quad S _ { n } < \frac { \pi } { 3 \sqrt { 3 } }$
(B) $\quad S _ { n } > \frac { \pi } { 3 \sqrt { 3 } }$
(C) $T _ { n } < \frac { \pi } { 3 \sqrt { 3 } }$
(D) $T _ { n } > \frac { \pi } { 3 \sqrt { 3 } }$

For $a \in \mathbb { R }$ (the set of all real numbers), $a \neq - 1$, $$\lim _ { \mathrm { n } \rightarrow \infty } \frac { \left( 1 ^ { a } + 2 ^ { a } + \ldots + \mathrm { n } ^ { a } \right) } { ( n + 1 ) ^ { a - 1 } [ ( n a + 1 ) + ( n a + 2 ) + \ldots + ( n a + n ) ] } = \frac { 1 } { 60 }$$ Then $a =$
(A) 5
(B) 7
(C) $\frac { - 15 } { 2 }$
(D) $\frac { - 17 } { 2 }$