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The powers of 2 form a sequence:
20  =   1
21  =   2
22  =   4
23  =   8
24  =  16
25  =  32
26  =  64
27  = 128

The sums of the rows in Pascal's triangle are equal to the powers of 2:
 

Sum of Row  ROW   

 

 

 

 

 

 

 

 

 1

 

 

 

 

 

 

 

 

1

 Row 0
 

 

 

 

 

 

 

 1

 

 1

 

 

 

 

 

 

 

2

 Row 1
 

 

 

 

 

 

 1

 

 2

 

 1

 

 

 

 

 

 

4

 Row 2
 

 

 

 

 

  1

 

 3

 

 3

 

 1

 

 

 

 

 

8

 Row 3
 

 

 

 

 1

 

 4

 

 6

 

 4

 

 1

 

 

 

 

16

 Row 4
 

 

 

 1

 

 5

 

10

 

10

 

 5

 

 1

 

 

 

32

 Row 5
 

 

 1

 

 6

 

15

 

20

 

15

 

 6

 

 1

 

 

64

 Row 6
 

 1

 

 7

 

21

 

35

 

35

 

21

 

 7

 

 1

 

128

 Row 7
 1

 

 8

 

28

 

56

 

70

 

56

 

28

 

 8

 

 1

   Row 8


 
The powers of 11 also form a sequence:

110  =   1
111  =   11
112  =   121
113  =   1331
114  =   14641
115  =   161051
116  =   1771561
117  =   19487171

The powers of 11 can be extracted from Pascal's triangle by reading across the rows and interpreting the digits as a place value system. Starting in row 5 the pattern becomes harder to see, because a two-digit number like the number 10 can not occupy a single place. You can think of row 5 in this way:

   1(105) + 5(104) + 10(103) + 10(102) + 5(101) + 1(100)

= 100000 + 50000 + 10000 + 1000 + 50 + 1

= 161051

 

Simplification of Row  ROW  

 

 

 

 

 

 

 

 

 1

 

 

 

 

 

 

 

 

1

 Row 0
 

 

 

 

 

 

 

 1

 

 1

 

 

 

 

 

 

 

11

 Row 1
 

 

 

 

 

 

 1

 

 2

 

 1

 

 

 

 

 

 

121

 Row 2
 

 

 

 

 

  1

 

 3

 

 3

 

 1

 

 

 

 

 

1331

 Row 3
 

 

 

 

 1

 

 4

 

 6

 

 4

 

 1

 

 

 

 

14641

 Row 4
 

 

 

 1

 

 5

 

10

 

10

 

 5

 

 1

 

 

 

161051

 Row 5
 

 

 1

 

 6

 

15

 

20

 

15

 

 6

 

 1

 

 

1771561

 Row 6
 

 1

 

 7

 

21

 

35

 

35

 

21

 

 7

 

 1

 

19487171

 Row 7
 1

 

 8

 

28

 

56

 

70

 

56

 

28

 

 8

 

 1

   Row 8
 

Let's look closely at Row 6:

 1

 6

15

20

 

15

 

 6

1

  1771561

 Row 6

 1

 

(6+1)

 

(5+2)

 

(0+1)

5

 

 6

1

   

 1

 7

 

 7

 

 1

 

5

 

 6

1

  1771561

 

 
And here's another way to think about the powers of 11:

    11 = (10+1)
Let x = 10

11^2

= (x+1)^2 = 1x^2 + 2x + 1 = 100 + 20 + 1 =  121

11^3

= (x+1)^3 = 1x^3 + 3x^2 + 3x + 1 = 1000 + 300 + 30 + 1 = 1331

11^4

= (x+1)^4 = 1x^4 + 4x^3 + 6x^2 + 4x + 1 = ...    
References
  1. Ask Dr. Math Archives: Exponents
  2. Eric Weisstein's World of Mathematics: Power


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Math Forum * * * * 21 January 2000