Download Pascal`s Triangle (answers)

Grade 11AP Mathematics
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Pascal’s Triangle (answers)
Date:
Pascal’s triangle is a triangular array of numbers that has numerous
applications in mathematics. The triangle is named for the great French
mathematician Blaise Pascal (1623 – 1662); however, there is evidence
that a famous Chinese algebraist, Chu Shi-kie, was familiar with the
number pattern as early as 1303. In a Pascal’s triangle, each term is equal
to the sum of the two terms immediately above it.
Draw at least the first 10 rows of Pascal’s Triangle.
Chu Shi-kie’s triangle
1
1
1
1
1
1
1
1
1
1
1
9
10
RHHS Mathematics Department
5
7
8
28
3
10
56
10
70
5
1
6
21
56
126
252
1
15
35
126
210
1
4
20
35
84
120
1
6
15
21
36
45
3
4
6
1
2
7
28
84
210
1
1
8
36
120
1
9
45
1
10
1
Grade 11AP Mathematics
Page 2 of 4
Pascal’s Triangle (answers)
Date:
Example 1
The first 5 terms in row 13 of Pascal’s triangle are 1, 13, 78, 286 and 715. Determine the first 5 terms in
row 14.
1
13
1
14
78
91
286
364
715
1001
Example 2
Shown below are portion of Pascal’s triangle. Fill in the missing numbers.
a)
b)
84
126
5
10
20
15
35
6
21
120
210
330
252
462
Example 3
a) Find the sums of the numbers in each of the first 6 rows of the Pascal’s
triangle and list them in a table of values.
n
∑t
Row, r
k ,r
k =0
b) Predict the sum of the entries in row 7.
0
1
c) Determine the row number from a Pascal’s triangle if the row sum is
2048.
1
2
211 = 2048 Therefore, it’s row 11 (assuming we start at row 0)
2
4
3
8
4
16
5
32
6
64
th
d) Determine a general rule for the sum of the terms in the n row.
n
∑ t n ,k = 2 n
k =0
Example 4
On the checkerboard shown, the checker can travel only diagonal upward. It
cannot move through a square containing an X. Determine the number of paths
from the checker’s current position to the top of the board.
5
9
5
1
30 ways to move the checker to the top of the board.
8
4
4
1
4
3
4
4
X
1
3
2
1
1
1
1
O
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8
Grade 11AP Mathematics
Page 3 of 4
Pascal’s Triangle (answers)
Date:
1. Shown below are portions of Pascal’s triangle. Fill in the missing numbers
a)
35
35
e)
b)
15
20
21
70
56
105
455
120
560
680
1820
2420
113
91
f)
14
1
105
15
18
153
171
19
190
c)
1
120
165
969
3003
715
2002
1001
364
1365
1716
3432
6435
286
3003
792
1716
1140
462
330
495
g)
d)
210
3003
6435
2. Determine the sum of the terms in each of these rows in Pascal’s triangle
a) row 12
b) row 25
c) row 20
3. Determine the row number for each of the following row sums from Pascal’s triangle.
a) 256
b) 4096
c) 16384
d) 65536
4. Refer to Example 4, let’s assume the checker may jump over the X into the diagonally opposite square.
How many paths are there to the top of the board now?
5. Coins can be arranged in the shape of an equilateral triangle as shown. This sequence of numbers are
called triangular numbers; t1 = 1, t 2 = 3, t 3 = 6, L .
a) Continue the pattern to determine the fourth, fifth and sixth triangular numbers.
b) Relate the sequence of triangular numbers to Pascal’s Triangle. State the nth triangular number in
terms of a Pascal’s Triangle term, tn,r.
c) Determine a general rule for the nth triangular number.
d) Determine the twelfth triangular number.
6. Matches are laid out to form triangles as shown.
a) Determine a general rule for the nth number in this matches sequence.
b) How many matches would the 10th triangle contain?
7. Spheres can be piled in a tetrahedral shape as shown on the right. Consequently,
this sequence is called tetrahedral numbers.
a) State the next 4 tetrahedral numbers, continuing the pattern:
t1 = 1, t 2 = 4, t 3 = _______, t 4 = _______, t 5 = _______, t 6 = _______, L
b) Relate the sequence of tetrahedral numbers to Pascal’s Triangle.
c) Determine a general rule for the nth tetrahedral number.
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Grade 11AP Mathematics
Page 4 of 4
Pascal’s Triangle (answers)
Date:
Answers
2a) 4096
b) 33554432
4. 5 + 12 + 14 + 11 = 42
6a)
3n(n + 1)
2
b) 165
c) 1048576
5b) r = 2; t n +1, 2
3a) 8
c)
b) 12
c) 14
n(n + 1)
2
d) 78
7a) 10, 20, 35, 56 b) r = 3; t n +1, 3
c)
d) 16
n(n + 1)(n + 2 )
6
Summary
•
•
•
Pascal’s triangle is symmetric
2n
The sum of the terms in row n of Pascal’s triangle is
Pascal’s Theorem generates the terms of Pascal’s triangle recursively:
 n   n   n + 1
 = 
 , where t n ,r represents the rth term in row n.
t n ,r + t n,r +1 = t n +1, r +1 or   + 
 r   r + 1  r + 1 
•
Triangular numbers:
The nth triangular number, t n =
•
“Matches” numbers:
tn =
•
3n(n + 1)
2
Tetrahedral numbers:
tn =
•
n(n + 1)
2
n(n + 1)(n + 2 )
6
n
General term of Pascal’s triangle, t n ,r =   = n C r where t n ,r represents the rth term in row n.
r
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