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© Boardworks Ltd of 39 Contents © Boardworks Ltd of 39 Binomial expansions Geometric sequences Geometric series The sum to infinity of a geometric series Binomial expansions Examination-style questions
© Boardworks Ltd of 39 Pascal’s Triangle
© Boardworks Ltd of 39 Binomial expansions An expression containing two terms, for example a + b, is called a binomial expression. When we find powers of binomial expressions an interesting pattern emerges. ( a + b ) 0 = 1 ( a + b ) 1 = 1 a + 1 b ( a + b ) 2 = 1 a ab + 1 b 2 ( a + b ) 3 = 1 a a 2 b + 3 ab b 3 ( a + b ) 4 = 1 a a 3 b + 6 a 2 b ab b 4 ( a + b ) 5 = 1 a a 4 b + 10 a 3 b a 2 b ab b 5 What patterns do you notice?
© Boardworks Ltd of 39 Binomial expansions In general, in the expansion of ( a + b ) n : The coefficients are given by the ( n + 1) th row of Pascal’s triangle. The sum of the powers of a and b is n for each term. Altogether, there are n + 1 terms in the expansion. As long as n is relatively small, we can expand a given binomial directly by comparing it to the equivalent expansion of ( a + b ) n. For example: Expand ( x + 1) 5. and replacing a with x and b with 1 gives: Using ( a + b ) 5 = a a 4 b + 10 a 3 b a 2 b ab 4 + b 5 ( x + 1) 5 = x x x x x + 1
© Boardworks Ltd of 39 Binomial expansions Expand (2 x – y ) 4. and replacing a with 2 x and b with – y gives: Using ( a + b ) 4 = a a 3 b + 6 a 2 b ab 3 + b 4 (2 x – y ) 4 = (2 x ) 4 + 4(2 x ) 3 (– y ) + 6(2 x ) 2 (– y ) 2 + 4(2 x )(– y ) 3 + (– y ) 4 = 16 x 4 – 32 x 3 y + 24 x 2 y 2 – 8 xy 3 + y 4 Notice that when the second term in a binomial is negative the signs of the terms in the expansion will alternate. Suppose we wanted to expand ( a + b ) 20. We could find the 21 st row of Pascal’s triangle, but this would take a very long time.
© Boardworks Ltd of 39 Finding binomial coefficients When n is large we can find the binomial coefficients using combinations theory. Let’s look more closely at the expansion of ( a + b ) 4 = ( a + b )( a + b )( a + b )( a + b ) aaaa 1 way aaab aaba abaa baaa 4 ways6 ways abbb babb bbab bbba 4 ways bbbb 1 way aabb abab abba bbaa baba baab Ways of getting a 4 Ways of getting a 3 b Ways of getting a 2 b 2 Ways of getting ab 3 Ways of getting b 4
© Boardworks Ltd of 39 Finding binomial coefficients The situation where no b ’s (or four a ’s) are chosen from any of the four brackets can be written as The situation where one b (or three a ’s) can be chosen from any of the four brackets can be written as: 4 C 0 or. The situation where two b ’s (or two a ’s) can be chosen from any of the four brackets can be written as 4 C 1 or. 4 C 2 or. This is the same as 4 C 4 or. This is the same as 4 C 3 or.
© Boardworks Ltd of 39 Finding binomial coefficients The fifth row of Pascal’s triangle can be written as: This corresponds to the values The expansion of ( a + b ) 4 can therefore be written as: Or:( a + b ) 4 = a a 3 b + 6 a 2 b ab 3 + b 4
© Boardworks Ltd of 39 Finding binomial coefficients The number of ways to choose r objects from a group of n objects is written as n C r and is given by n ! is read as ‘ n factorial’ and is the product of all the natural numbers from 1 to n. In general: n ! = n × ( n –1) × ( n – 2) × ( n – 3) … × 2 × 1 n can also be 0 and by definition 0! = 1.
© Boardworks Ltd of 39 Finding binomial coefficients The value of n! gets large very quickly as the value of n increases. For example: 5! = 5 × 4 × 3 × 2 × 1 =120 12! = 12 × 11 × 10 × … × 2 × 1 = Fortunately, when we use the formula to calculate binomial coefficients, many of the numbers cancel out. For example, for 4 C 2 we have 20! = 20 × 19 × 18 × … × 2 × 1 = = 6 2
© Boardworks Ltd of 39 Finding binomial coefficients Here are some more examples: 56 This value corresponds to the number of ways of choosing 3 a ’s from the 8 brackets in the expansion of ( a + b ) is therefore the coefficient of a 3 b 5 in the expansion of ( a + b ) This value corresponds to the number of ways of choosing 7 a ’s from the 9 brackets in the expansion of ( a + b ) is therefore the coefficient of a 7 b 2 in the expansion of ( a + b ) 9. 4
© Boardworks Ltd of 39 Finding binomial coefficients The effect of this cancelling gives an alternative form for n C r. In general, the expansion of ( a + b ) n can be written as: A special case is the expansion of (1 + x ) n
© Boardworks Ltd of 39 Using the binomial theorem Find the coefficient of a 7 b 3 in the expansion of ( a – 2 b ) 10. This method of finding the binomial coefficients is called the binomial theorem. The term in a 7 b 3 is of the form: 4 = 120(–8 a 7 b 3 ) So the coefficient of a 7 b 3 in the expansion of ( a – 2 b ) 10 is –960. = –960 a 7 b 3 3
© Boardworks Ltd of 39 Using the binomial theorem Use the binomial theorem to write down the first four terms in the expansion of (1 + x ) 7 in ascending powers of x. 3 How could we use this expansion to find an approximate value for ?
© Boardworks Ltd of 39 Using the binomial theorem To find an approximate value for we can let x = 0.1 in the expansion (1 + x ) 7 = x + 21 x x 3 + … This gives us We can therefore leave out higher powers of x and still have a reasonable approximation ≈ As 0.1 is raised to ever higher powers it becomes much smaller and so less significant ≈ 1 +7 × × × ≈ 1.945
© Boardworks Ltd of 39 Contents © Boardworks Ltd of 39 Examination-style questions Geometric sequences Geometric series The sum to infinity of a geometric series Binomial expansions Examination-style questions
© Boardworks Ltd of 39 Examination-style question 2 a)Write down the first four terms in the expansion of (1 + ax ) 13 in ascending powers of x, where a > 0. b)Given that in the expansion of (1 + ax ) 13 the coefficient of x is – b and the coefficient of x 2 is 12 b, find the value of a and b. a) (1 + ax ) 13 = = ax + 78 a 2 x a 3 x 3 + … b)13 a = – b 1 78 a 2 = 12 b 2 78 a 2 = 12 × –13 a Substituting into : a = –156 a = –2 b = 26
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