902 CHAPTER 12 Sequences; Induction; the Binomial Theorem As this solution demonstrates, the Binomial Theorem can be used to find a particular term in the expansion of ( ) + ax b n without writing the entire expansion. Example 4 can be solved by using formula (3) with = = = n a b 10, 2, 3, and = j 8. Then the term containing y8 is ( ) − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ ⋅ ⋅ ⋅ = ⋅ ⋅ = ⋅ ⋅ ⋅ ⋅ ⋅ = − y y y y y 10 10 8 3 2 10 2 3 2 10! 2! 8! 9 2 10 9 8! 2 8! 9 2 103,680 10 8 8 2 8 8 8 8 8 8 8 The term containing xj in the expansion of ( ) + ax b n is ( ) − ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ − n n j b ax n j j (3) The 6th term in the expansion of ( ) +x 2 ,9 which has 10 terms total, contains x .4 (Do you see why?) By formula (3), the 6th term is − ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ = ⋅ = − x x x x 9 9 4 2 9 5 2 9! 5! 4! 32 4032 9 4 4 5 4 4 4 Finding a Particular Term in a Binomial Expansion Find the 6th term in the expansion of ( ) +x 2 . 9 Solution A EXAMPLE 5 Expand using the Binomial Theorem until the 6th term is reached. ( ) + = ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ ⋅ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ ⋅ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ ⋅ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ ⋅ + ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ ⋅ + x x x x x x x 2 9 0 9 1 2 9 2 2 9 3 2 9 4 2 9 5 2 9 9 8 7 2 6 3 5 4 4 5 The 6th term is ⎛ ⎝ ⎜⎜ ⎜ ⎞ ⎠ ⎟⎟ ⎟ ⋅ = ⋅ ⋅ = x x x 9 5 2 9! 5! 4! 32 4032 4 5 4 4 Solution B Now Work PROBLEMS 29 AND 35 The following theorem shows that the triangular addition feature of the Pascal triangle illustrated in Figure 22 always works. THEOREM If n and j are integers with ≤ ≤ j n 1 , then − ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ + ⎛ ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ = ⎛ + ⎝ ⎜⎜ ⎜⎜ ⎞ ⎠ ⎟⎟ ⎟⎟ n j n j n j 1 1 (4)
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