900 CHAPTER 12 Sequences; Induction; the Binomial Theorem Figure 22 The Pascal triangle 1 5 10 10 5 1 4 6 4 1 3 3 1 2 1 1 n = 0 n = 1 n = 2 n = 3 n = 4 n = 5 1 1 1 1 1 j 5 5 j 5 4 j 5 3 j 5 2 j 5 1 j 5 0 Suppose that the values of the symbol ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j are arranged in a triangular display, as shown next and in Figure 22. ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) 0 0 1 0 1 1 2 0 2 1 2 2 3 0 3 1 3 2 3 3 4 0 4 1 4 2 4 3 4 4 5 0 5 1 5 2 5 3 5 4 5 5 This display is called the Pascal triangle , named after Blaise Pascal (1623–1662), a French mathematician. The Pascal triangle has 1’s down the sides. To get any other entry, add the two nearest entries in the row above it. The shaded triangles in Figure 22 illustrate this feature of the Pascal triangle. Based on this feature, the row corresponding to = n 6 is found as follows: = → = → 1 5 10 10 5 1 16 15201561 n n 5 6 This addition always works (see the theorem on page 902). Although the Pascal triangle provides an interesting and organized display of the symbol ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j , in practice it is not that helpful. For example, if you want the value of ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ 12 5 , you would need 13 rows of the triangle before seeing the answer. It is much faster to use definition (1). 2 Use the Binomial Theorem NOTE The Pascal triangle is symmetric about a line down its center. If we divide the triangle vertically in half, the entries in each row of the left half are the mirror image of the entries in the same row of the right half. 1 5 10 10 5 1 1 4 6 4 1 1 3 3 1 1 2 1 1 1 1 The vertical symmetry of the entries in the Pascal triangle is a result of the fact that ( ) ( ) − ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟⎟ = − = − = ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟⎟ n n j n n j j n j n j n j ! ! ! ! ! ! j THEOREM Binomial Theorem Let x and a be real numbers. For any positive integer n , ∑ ( ) + = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + + ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ + + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − − = − x a n x n ax n j a x n n a n j a x 0 1 n n n j n j n j n j n j 1 0 (2)
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