Let us define a sequence $\left\{ S _ { n } \right\}$ as $$S _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { \sqrt { k } } \quad ( n = 1,2,3 , \cdots ) .$$ We are to find the following two limits: $$\begin{aligned}
& \lim _ { n \rightarrow \infty } S _ { n } , \\
& \lim _ { n \rightarrow \infty } \frac { S _ { 2 n } - S _ { n } } { \sqrt { n } } .
\end{aligned}$$ (1) For each of A $\sim$ I in the following sentences, choose the appropriate answer from among (0) $\sim$ (9) at the bottom of this page. Let us find $\lim _ { n \rightarrow \infty } S _ { n }$. Look at the function $y = \frac { 1 } { \sqrt { x } }$. We have $$y ^ { \prime } = - \frac { \mathbf { A } } { 2 \sqrt { x ^ { \mathbf { B} } } } ,$$ and hence this function $y$ is $\square$ C . So, considering each interval $k \leqq x \leqq k + 1 ( k = 1,2 , \cdots , n )$, we obtain $$\frac { 1 } { \sqrt { k } } \mathbf { D } \int _ { k } ^ { k + 1 } \frac { 1 } { \sqrt { x } } d x .$$ When we separately add the left-hand sides and the right-hand sides of this expression from $k = 1$ to $k = n$, we have $$S _ { n } \mathbf { E } \int _ { \mathbf { F } } ^ { \mathbf { G } } \frac { 1 } { \sqrt { x } } d x = \mathbf { H } ( \sqrt { \square \mathbf { G } } - 1 )$$ and finally $$\lim _ { n \rightarrow \infty } S _ { n } = \infty .$$ Choices: (0) $\infty$ (1) 1 (2) 2 (3) 3 (4) $n$ (5) $n + 1$ (6) $<$ (7) $>$ (8) monotonically increasing (9) monotonically decreasing (2) For each of $\square$ J $\sim$ $\square$ P in the following, choose the appropriate answer from among (0) $\sim$ (9) below. Let us find $\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } - S _ { n } } { \sqrt { n } }$. Since $$S _ { 2 n } - S _ { n } = \sum _ { k = 1 } ^ { n } \frac { 1 } { \sqrt { \mathbf { J } } } ,$$ we have from quadrature (mensuration) by parts that $$\begin{aligned}
\lim _ { n \rightarrow \infty } \frac { S _ { 2 n } - S _ { n } } { \sqrt { n } } & = \lim _ { n \rightarrow \infty } \frac { 1 } { \mathbf { K } } \sum _ { k = 1 } ^ { n } \frac { 1 } { \sqrt { \mathbf { L } + \frac { k } { n } } } \\
& = \int _ { \mathbf { M } } ^ { \mathbf { N } } \frac { 1 } { \sqrt { 1 + x } } d x \\
& = \mathbf { O } ( \sqrt { \mathbf { P } } - 1 ) .
\end{aligned}$$ Choices: (0) 0 (1) 1 (2) 2 (3) $n - 1$ (4) $n$ (5) $n + 1$ (6) $n - k$ (7) $n + k$ (8) $n + k - 1$ (9) $n + k + 1$
This question concerns the sum $s _ { n }$ defined by $$s _ { n } = 2 + 8 + 24 + \cdots + n 2 ^ { n }$$ (i) Let $f ( n ) = ( A n + B ) 2 ^ { n } + C$ for constants $A , B$ and $C$ yet to be determined, and suppose $s _ { n } = f ( n )$ for all $n \geqslant 1$. By setting $n = 1,2,3$, find three equations that must be satisfied by $A , B$ and $C$. (ii) Solve the equations from part (i) to obtain values for $A , B$ and $C$. (iii) Using these values, show that if $s _ { k } = f ( k )$ for some $k \geqslant 1$ then $s _ { k + 1 } = f ( k + 1 )$. You may now assume that $f ( n ) = s _ { n }$ for all $n \geqslant 1$. (iv) Find simplified expressions for the following sums: $$\begin{aligned}
& t _ { n } = n + 2 ( n - 1 ) + 4 ( n - 2 ) + 8 ( n - 3 ) + \cdots + 2 ^ { n - 1 } 1 , \\
& u _ { n } = \frac { 1 } { 2 } + \frac { 2 } { 4 } + \frac { 3 } { 8 } + \cdots + \frac { n } { 2 ^ { n } } .
\end{aligned}$$ (v) Find the sum $$\sum _ { k = 1 } ^ { n } s _ { k }$$ If you require additional space please use the pages at the end of the booklet
There is a unique real number $\alpha$ that satisfies the equation $$\alpha ^ { 3 } + \alpha ^ { 2 } = 1$$ [You are not asked to prove this.] (i) Show that $0 < \alpha < 1$. (ii) Show that $$\alpha ^ { 4 } = - 1 + \alpha + \alpha ^ { 2 }$$ (iii) Four functions of $\alpha$ are given in (a) to (d) below. In a similar manner to part (ii), each is equal to a quadratic expression $$A + B \alpha + C \alpha ^ { 2 }$$ in $\alpha$, where $A , B , C$ are integers. (So in (ii) we found $A = - 1 , B = 1 , C = 1$.) You may assume in each case that the quadratic expression is unique. In each case below find the quadratic expression in $\alpha$. (a) $\alpha ^ { - 1 }$. (b) The infinite sum $$1 - \alpha + \alpha ^ { 2 } - \alpha ^ { 3 } + \alpha ^ { 4 } - \alpha ^ { 5 } + \cdots$$ (c) $( 1 - \alpha ) ^ { - 1 }$. (d) The infinite product $$( 1 + \alpha ) \left( 1 + \alpha ^ { 2 } \right) \left( 1 + \alpha ^ { 4 } \right) \left( 1 + \alpha ^ { 8 } \right) \left( 1 + \alpha ^ { 16 } \right) \cdots$$
Let $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$ be a geometric sequence with first term 3 and common ratio $3\sqrt{3}$. Select the number of terms $n$ that satisfy the inequality $$\log_{3} a_{1} - \log_{3} a_{2} + \log_{3} a_{3} - \log_{3} a_{4} + \ldots + (-1)^{n+1} \log_{3} a_{n} > 18$$ among the possible options. (1) 23 (2) 24 (3) 25 (4) 26 (5) 27
16. The sequence $a _ { n }$ is given by the rule: $$\begin{aligned}
a _ { 1 } & = 2 \\
a _ { n + 1 } & = a _ { n } + ( - 1 ) ^ { n } \text { for } n \geq 1
\end{aligned}$$ What is $$\sum _ { n = 1 } ^ { 100 } a _ { n }$$ A 150 B 250 C - 4750 D 5150 E $\quad 4 \left( 1 - \left( \frac { 1 } { 2 } \right) ^ { 100 } \right)$ F $\quad 4 \left( \left( \frac { 3 } { 2 } \right) ^ { 100 } - 1 \right)$
For a sequence $a_n$ where the sum of any three consecutive terms is equal to each other, $$a _ { 2 } + a _ { 3 } = a _ { 4 } = 2$$ equality is satisfied. Accordingly, $$a _ { 1 } + a _ { 2 } + \ldots + a _ { 25 }$$ what is the result of the sum? A) 34 B) 35 C) 36 D) 37 E) 38