Let $S _ { K } = \frac { 1 + 2 + \ldots + K } { K }$ and $\sum _ { j = 1 } ^ { n } S ^ { 2 } { } _ { j } = \frac { n } { A } \left( B n ^ { 2 } + C n + D \right)$ where $A , B , C , D \in N$ and $A$ has least value then
(1) $A + C + D$ is not divisible by $D$
(2) $A + B = 5 ( D - C )$
(3) $A + B + C + D$ is divisible by 5
(4) $A + B$ is divisible by $D$

If $\sum _ { r = 1 } ^ { n } T _ { r } = \frac { ( 2 n - 1 ) ( 2 n + 1 ) ( 2 n + 3 ) ( 2 n + 5 ) } { 64 }$, then $\lim _ { n \rightarrow \infty } \sum _ { r = 1 } ^ { n } \left( \frac { 1 } { T _ { r } } \right)$ is equal to:
(1) 0
(2) $\frac { 2 } { 3 }$
(3) 1
(4) $\frac { 1 } { 3 }$

Q82. Let the positive integers be written in the form : [Figure]
If the $k ^ { \text {th } }$ row contains exactly $k$ numbers for every natural number $k$, then the row in which the number 5310 will be, is $\_\_\_\_$

Find the value of $\sum _ { k = 1 } ^ { \infty } \frac { ( - 1 ) ^ { \mathbf { k } + \mathbf { 1 } } \cdot \mathbf { k } ( \mathbf { k } + \mathbf { 1 } ) } { \mathbf { k } ! }$ (A) $\frac { 2 } { e }$ (B) $\frac { 3 } { e }$ (C) $\frac { 1 } { e }$ (D) e

5. For ALL APPLICANTS.

Let $n$ be a positive integer. An $n$-brick is a rectangle of height 1 and width $n$. A 1 -tower is defined as a 1 -brick. An $n$-tower, for $n \geqslant 2$, is defined as an $n$-brick on top of which exactly two other towers are stacked: a $k _ { 1 }$-tower and a $k _ { 2 }$-tower such that $1 \leqslant k _ { 1 } \leqslant n - 1$ and $k _ { 1 } + k _ { 2 } = n$. The $k _ { 1 }$-tower is placed to the left of the $k _ { 2 }$-tower so that side-by-side they fit exactly on top of the $n$-brick. For example, here is a 4 -tower: [Figure]
(i) Draw the four other 4 -towers.
(ii) What is the maximum height of an $n$-tower? Justify your answer.
(iii) The area of a tower is defined as the sum of the widths of its bricks. For example, the 4 -tower drawn above has area $4 + 4 + 3 + 2 = 13$. Give an expression for the area of an $n$-tower of maximum height.
(iv) Show that there are infinitely many $n$ such that there is an $n$-tower of height exactly $1 + \log _ { 2 } n$.
(v) Write $t _ { n }$ for the number of $n$-towers. We have $t _ { 1 } = 1$. For $n \geqslant 2$ give a formula for $t _ { n }$ in terms of $t _ { k }$ for $k < n$. Use your formula to compute $t _ { 6 }$.
(vi) Show that $t _ { n }$ is odd if and only if $t _ { 2 n }$ is odd.