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Sequence - Finding a Pattern

SOURCE:COMPETITION
Number of Problems: 10.
FOR PRINT ::: (Book)

Problem Num : 1

Type: None

Topic:Sequence

Theme:None
Adjustment# :

Difficulty: 1

Category Finding a Pattern

Analysis

Solution/Hint

Problem Num : 2

Type: None

Topic:Sequence

Theme:None
Adjustment# :

Difficulty: 1

Category Finding a Pattern

Analysis

Solution/Hint

Problem Num : 3

Type: Application

Topic:Sequence

Theme:Pattern
Adjustment# :

Difficulty: 1

Category Finding a Pattern

Analysis

Solution/Hint

Problem Num : 4

Type: None

Topic:Sequence

Theme:None
Adjustment# :

Difficulty: 1

Category Finding a Pattern

Analysis

Solution/Hint

Problem Num : 5

Type: None

Topic:Sequence

Theme:Pattern
Adjustment# :

Difficulty: 2

Category Finding a Pattern

Analysis

Solution/Hint

Problem Num : 6

Type: None

Topic:Sequence

Theme:Manipulation
Adjustment# :

Difficulty: 1

Category Finding a Pattern

Analysis

Solution/Hint

Problem Num : 7

Type:

Topic:Sequence

Adjustment# : 0

Difficulty: 1

'
A grocer stacks oranges in a pyramid-like stack whose rectangular base is 5 oranges by 8 oranges. Each orange above the first level rests in a pocket formed by four oranges below. The stack is completed by a single row of oranges. How many oranges are in the stack?
$mathrm{(A) } 96 qquad mathrm{(B) } 98 qquad mathrm{(C) } 100 qquad mathrm{(D) } 101 qquad mathrm{(E) } 134$
'

Category Finding a Pattern

Analysis

Solution/Hint
There are $5 imes8=40$ oranges on the 1st layer of the stack. When the 2nd layer is added on top of the first, it will be a layer of $4 imes7=28$ oranges. When the third layer is added on top of the 2nd, it will be a layer of $3 imes6=18$ oranges, etc.
Therefore, there are $5 imes8+4 imes7+3 imes6+2 imes5+1 imes4=40+28+18+10+4=100$ oranges in the stack $Rightarrowmathrm{(C)}$ .

Problem Num : 8

Type:

Topic:Sequence

Adjustment# : 0

Difficulty: 1

'
Let $f$ be a function with the following properties:

$(i)quad f(1) = 1$ , and
$(ii)quad f(2n) = n imes f(n)$ , for any positive integer $n$ .

What is the value of $f(2^{100})$ ?
$ext {(A)} 1 qquad ext {(B)} 2^{99} qquad ext {(C)} 2^{100} qquad ext {(D)} 2^{4950} qquad ext {(E)} 2^{999...$
'

Category Finding a Pattern

Analysis

Solution/Hint
$f(2^{100}) = f(2 imes 2^{99}) = 2^{99} imes f(2^{99})$ $= 2^{99} cdot 2^{98} imes f(2^{98}) = ldots$ $= 2^{99}2^{98}cdots 2^{1} cdot 1 cdot f(1)$ $= 2^{99 + 98 + ldots + 2 + 1}$ $= 2^{frac{99(100)}{2}} = 2^{4950}$ $Rightarrow mathrm{(D)}$ .

Problem Num : 9

Type: Understanding

Topic:Sequence

Theme:Pattern
Adjustment# : 0

Difficulty: 2

Category Finding a Pattern

Analysis

Solution/Hint

Problem Num : 10

Type: Understanding

Topic:Sequence

Theme:Pattern
Adjustment# : 2

Difficulty: 3

Category Finding a Pattern

Analysis ?????? ?? ?? ?????? ???????? ?????? ?????? ??? ???? ???????.

Solution/Hint