Two sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ are given by $$\begin{aligned} & a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } \cos \frac { ( n - 1 ) \pi } { 2 } \\ & b _ { n } = \frac { 1 + ( - 1 ) ^ { n - 1 } } { 2 ^ { n } } \end{aligned}$$ Which of the following in are correct? [4 points] 〈Remarks〉 ㄱ. For all natural numbers $k$, $a _ { 3 k } < 0$. ㄴ. For all natural numbers $k$, $a _ { 4 k - 1 } + b _ { 4 k - 1 } = 0$. ㄷ. $\sum _ { n = 1 } ^ { \infty } a _ { n } = \frac { 3 } { 5 } \sum _ { n = 1 } ^ { \infty } b _ { n }$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

Two sequences $\left\{ a _ { n } \right\} , \left\{ b _ { n } \right\}$ are given by
$$\begin{aligned} & a _ { n } = \frac { 1 } { 2 ^ { n - 1 } } \cos \frac { ( n - 1 ) \pi } { 2 } \\ & b _ { n } = \frac { 1 + ( - 1 ) ^ { n - 1 } } { 2 ^ { n } } \end{aligned}$$
Which of the following statements in are true? [4 points]

ㄱ. For all natural numbers $k$, $a _ { 3 k } < 0$. ㄴ. For all natural numbers $k$, $a _ { 4 k - 1 } + b _ { 4 k - 1 } = 0$. ㄷ. $\sum _ { n = 1 } ^ { \infty } a _ { n } = \frac { 3 } { 5 } \sum _ { n = 1 } ^ { \infty } b _ { n }$
(1) ㄱ
(2) ㄴ
(3) ㄷ
(4) ㄱ, ㄴ
(5) ㄴ, ㄷ

6. Let $\left\{ a _ { n } \right\}$ be an arithmetic sequence. The correct conclusion is
A. If $a _ { 1 } + a _ { 2 } > 0$, then $a _ { 2 } + a _ { 3 } > 0$
B. If $a _ { 1 } + a _ { 3 } < 0$, then $a _ { 1 } + a _ { 2 } < 0$
C. If $0 < a _ { 1 } < a _ { 2 }$, then $a _ { 2 } > \sqrt { a _ { 1 } a _ { 3 } }$
D. If $a _ { 1 } < 0$, then $\left( a _ { 2 } - a _ { 1 } \right) \left( a _ { 2 } - a _ { 3 } \right) > 0$

20. Let $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ be terms of an arithmetic sequence with positive terms and common difference $\mathrm { d } ( d \neq 0 )$
(1) Prove that $2 ^ { a _ { 1 } } , 2 ^ { a _ { 2 } } , 2 ^ { a _ { 3 } } , 2 ^ { a _ { 4 } }$ form a geometric sequence in order
(2) Do there exist $a _ { 1 } , d$ such that $a _ { 1 } , a _ { 2 } { } ^ { 2 } , a _ { 3 } { } ^ { 3 } , a _ { 4 } { } ^ { 4 }$ form a geometric sequence in order? Explain your reasoning
(3) Do there exist $a _ { 1 } , d$ and positive integers $n , k$ such that $a _ { 1 } { } ^ { n } , a _ { 2 } { } ^ { n + k } , a _ { 3 } { } ^ { n + 3 k } , a _ { 4 } { } ^ { n + 5 k }$ form a geometric sequence in order? Explain your reasoning

Supplementary Problems

Given that the arithmetic sequence $\left\{ a _ { n } \right\}$ has common difference $\frac { 2 \pi } { 3 }$, and the set $S = \left\{ \cos a _ { n } \mid n \in \mathbb{N} ^ { * } \right\}$. If $S = \{ a , b \}$, then $a b =$
A. $- 1$
B. $- \frac { 1 } { 2 }$
C. 0
D. $\frac { 1 } { 2 }$

15. $\operatorname { Lim } x \rightarrow 1 \sqrt { } ( 1 - \cos 2 ( x - 1 ) ) / ( x - 1 )$ :
(A) exists and it equals $\sqrt { } 2$.
(B) exists and it equals $- \sqrt { } 2$
(C) does not exist because $x - 1 - - > 0$
(D) does not exist because left hand limit is not equal to right hand limit
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1. If in a triangle $P Q R , \sin P , \sin Q , \sin R$ are in $A$. $P$., then :
   (A) the altitudes are in A.P.
   (B) the altitudes are in H.P.
   (C) the medians are in G.P.
   (D) the medians are in A.P.
2. If $a n = \sum r = 0 n 1 / n \mathrm { Cr }$, then $\sum r = 0 n \mathrm { r } / \mathrm { n }$ Cr equals:
   (A) (n - 1) an
   (B) $n$ an
   (C) $1 / 2$ nan
   (D) none of these
3. If the vertices $P , Q , R$ of a triangle $P Q R$ are rational points, which of the following points of the triangle PQR is/(are) always rational point(s).
   (A) centroid \&
   (B) incentre
   (C) circumcentre
   (D) orthocenter
   (A rational point is a point both of whose co-ordinates are rational numbers).
4. The number of values of $c$ such that the straight line $y = 4 x + c$ touches the curve $x 2 / 4 + \mathrm { y } 2 = 1$ is :
   (A) 0
   (B) 1
   (C) 2
   (D) infinite.
5. If $x > 1 , y > 1 , z > 1$ are in G.P., then $1 / ( 1 + \operatorname { In } x ) , 1 / ( 1 + \operatorname { In } y ) , 1 / ( 1 + \operatorname { In } z )$ are in :
   (A) A.P.
   (B) H.P.
   (C) G.P.
   (D) none of these
6. The number of values of $x$ in the interval $[ 0,5 p ]$ satisfying the equation $3 \sin 2 x - 7 x + 2 = 0$ is:
   (A) 0
   (B) 5
   (C) 6
   (D) 10
7. The order of the differential equation whose general solution is given by

$$\begin{aligned} & y = \left( C _ { 1 } + C _ { 2 } \right) \cos \left( x + C _ { 3 } \right) - C _ { 4 } e ^ { x + C s } \\ & \text { where } C _ { 1 } , C _ { 2 } , C _ { 3 } , C _ { 4 } , C _ { 5 } \end{aligned}$$
are arbitrary constants, is:

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(A) 5
(B) 4
(C) 3
(D) 2

24. Let the positive numbers $\mathrm { a } , \mathrm { b } , \mathrm { c } , \mathrm { d }$ be in A.P. Then $\mathrm { abc } , \mathrm { abd } , \mathrm { acd } , \mathrm { bcd }$ are:

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(A) NOT in A.P./G.P/H.P.
(B) In A.P
(C) In G.P
(D) In H.P

If the A.M. between $p ^ { \text {th} }$ and $q ^ { \text {th} }$ terms of an A.P. is equal to the A.M. between $r ^ { \text {th} }$ and $s ^ { \text {th} }$ terms of the same A.P., then $p + q$ is equal to
(1) $r + s - 1$
(2) $r + s - 2$
(3) $r + s + 1$
(4) $r + s$

If $a, b, c, d$ and $p$ are distinct real numbers such that $\left(a^{2}+b^{2}+c^{2}\right)p^{2} - 2p(ab+bc+cd) + \left(b^{2}+c^{2}+d^{2}\right) \leq 0$, then
(1) $a, b, c, d$ are in A.P.
(2) $ab = cd$
(3) $ac = bd$
(4) $a, b, c, d$ are in G.P.

If $x, y, z$ are in AP and $\tan^{-1}x$, $\tan^{-1}y$ and $\tan^{-1}z$ are also in AP, then
(1) $x = y = z$
(2) $2x = 3y = 6z$
(3) $6x = 3y = 2z$
(4) $6x = 4y = 3z$

If $x, y, z$ are positive numbers in A.P. and $\tan^{-1}x, \tan^{-1}y$ and $\tan^{-1}z$ are also in A.P., then which of the following is correct.
(1) $6x = 3y = 2z$
(2) $6x = 4y = 3z$
(3) $x = y = z$
(4) $2x = 3y = 6z$

For any three positive real numbers $a$, $b$ and $c$, $9 ( 25 a ^ { 2 } + b ^ { 2 } ) + 25 ( c ^ { 2 } - 3 a c ) = 15 b ( 3 a + c )$. Then:
(1) $b$, $c$ and $a$ are in G.P.
(2) $b$, $c$ and $a$ are in A.P.
(3) $a$, $b$ and $c$ are in A.P.
(4) $a$, $b$ and $c$ are in G.P.

For any three positive real numbers $a, b$ and $c$. If $9(25a^2 + b^2) + 25(c^2 - 3ac) = 15b(3a + c)$. Then
(1) $b,\ c$ and $a$ are in G.P.
(2) $b,\ c$ and $a$ are in A.P.
(3) $a,\ b$ and $c$ are in A.P.
(4) $a,\ b$ and $c$ are in G.P.

If ${ } ^ { n } C _ { 4 } , { } ^ { n } C _ { 5 }$ and ${ } ^ { n } C _ { 6 }$ are in A.P., then $n$ can be
(1) 9
(2) 14
(3) 12
(4) 11

A function $f ( x )$ is given by $f ( x ) = \frac { 5 ^ { x } } { 5 ^ { x } + 5 }$, then the sum of the series $f \left( \frac { 1 } { 20 } \right) + f \left( \frac { 2 } { 20 } \right) + f \left( \frac { 3 } { 20 } \right) + \ldots + f \left( \frac { 39 } { 20 } \right)$ is equal to:
(1) $\frac { 19 } { 2 }$
(2) $\frac { 49 } { 2 }$
(3) $\frac { 39 } { 2 }$
(4) $\frac { 29 } { 2 }$

If $x = \sum _ { n = 0 } ^ { \infty } a ^ { n } , y = \sum _ { n = 0 } ^ { \infty } b ^ { n } , z = \sum _ { n = 0 } ^ { \infty } c ^ { n }$, where $a , b , c$ are in A.P. and $| a | < 1 , | b | < 1 , | c | < 1 , a b c \neq 0$, then
(1) $x , y , z$ are in A.P.
(2) $x , y , z$ are in G.P.
(3) $\frac { 1 } { x } , \frac { 1 } { y } , \frac { 1 } { z }$ are in A.P.
(4) $\frac { 1 } { x } + \frac { 1 } { y } + \frac { 1 } { z } = 1 - ( a + b + c )$

If $\log _ { e } a , \log _ { e } b , \log _ { e } c$ are in an $A . P$. and $\log _ { e } a - \log _ { e } 2 b , \log _ { e } 2 b - \log _ { e } 3 c , \log _ { e } 3 c - \log _ { e } a$ are also in an $A . P$. , then $a : b : c$ is equal to
(1) $9 : 6 : 4$
(2) $16 : 4 : 1$
(3) $25 : 10 : 4$
(4) $6 : 3 : 2$

8. Suppose real numbers $a _ { 1 } , a _ { 2 } , a _ { 3 } , a _ { 4 }$ form an arithmetic sequence, and satisfy $0 < a _ { 1 } < 2$ and $a _ { 3 } = 4$. If $b _ { n } = 2 ^ { a _ { n } }$ is defined, which of the following options are correct?
(1) $b _ { 1 } , b _ { 2 } , b _ { 3 } , b _ { 4 }$ form a geometric sequence.
(2) $b _ { 1 } < b _ { 2 }$.
(3) $b _ { 2 } > 4$.
(4) $b _ { 4 } > 32$.
(5) $b _ { 2 } \times b _ { 4 } = 256$.

Let the arithmetic sequence $\left\langle a _ { n } \right\rangle$ have first term $a _ { 1 }$ and common difference $d$ both positive, and $\log a _ { 1 } , \log a _ { 3 } , \log a _ { 6 }$ also form an arithmetic sequence in order. Select the common difference of the sequence $\log a _ { 1 } , \log a _ { 3 } , \log a _ { 6 }$.
(1) $\log d$
(2) $\log \frac { 2 } { 3 }$
(3) $\log \frac { 3 } { 2 }$
(4) $\log 2 d$
(5) $\log 3 d$

An arithmetic sequence has a first term of 1, a last term of 81, and 9 is also in the sequence. Let the number of terms in this sequence be $n$, where $n \leq 100$ . Select the correct options.
(1) $n$ is odd
(2) 41 must be in this arithmetic sequence
(3) The common difference of all arithmetic sequences satisfying the conditions is an integer
(4) There are 10 arithmetic sequences satisfying the conditions
(5) If $n$ is a multiple of 7, then $n = 21$

In this question, $x _ { 1 } , x _ { 2 } , x _ { 3 } , \ldots$ is an arithmetic progression, all of whose terms are integers.
Let $n$ be a positive integer. If the median of the first $n$ terms of the sequence is an integer, which of the following three statements must be true?
I The median of the first $n + 2$ terms is an integer.
II The median of the first $2 n$ terms is an integer.
III The median of $x _ { 2 } , x _ { 4 } , x _ { 6 } , \ldots , x _ { 2 n }$ is an integer.

An arithmetic series has $n$ terms, all of which are integers.
The sum of the series is 20 .
Which of the following statements must be true?
I The first term of the series is even.
II $n$ is even.
III The common difference is even.