A-Level Mathematics FACTORIALS and THE BINOMIAL THEOREM

G. David Boswell - RS2 Discovery 2018

SET A FACTORIALS: DEFINITIONS and ALGEBRA

Review ~ By definition, ! , then !

Also, in our proofs and simplifications of expressions, we often use manipulations such as: (i) !

(ii) !

A.3 Computations

Without using your calculators, find the exact value of each of the following.

1. !

2. !

n∈! n!=n(n −1)(n − 2)× ...× 2 ×1 , if n∈!

1 , if n = 0beyond scope , if n∉W

⎧

⎨⎪

⎩⎪

(n + 2)!= (n − 2)(n −1)n!

(n − 2)!= n(n −1)(n − 2)!n(n −1)

= n!n(n −1)

3!7!

6!9!7!3!

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A.3 Computations

Simplify the following expressions.

3. !

4. !

5. !

A.2 Proofs involving the use of the Factorial definitions

6. Show that ! where !

7. Show that ! . What is a most useful application of this

result?

8. Given that ! , show that ! is equal to the binomial coefficient ! .

9. Using the identity ! and given that ! , show that

!

(n − 2)!n2 (n +1)!

n!(n +1)!

p!(1+ q)!

(6q)!(p − 2)!

n−1Cr−1 +n−1Cr =

nCrnCr =

n!(n − r)!r!

nCr =n(n −1)(n − 2)× ...× (n − r +1)

r!

nPr =n!

(n − r)!

2rPrnPr

2r( )!nCr

1+ x( )2n = 1+ x( )n 1+ x( )n Cr =nr

⎛⎝⎜

⎞⎠⎟

2nn

⎛⎝⎜

⎞⎠⎟= C0

2 +C12 +C2

2 + ...+Cn−12 +Cn

2

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SET B THE BINOMIAL THEOREM and COEFFICIENTS

Review ~

(i) The Binomial Theorem for Positive Integral Index is stated as:

!

This series contains n+1 terms and is a finite series.

(ii) The Binomial Theorem for Negative or Fractional Index is an infinite series which is

equivalent to and commonly written as:

!

A more generalised form of the Binomial Theorem for Negative or Fractional index is:

!

In this case of non-positive index, the series converge if and only if ! .

(iii) Two very Special Cases of the Binomial Theorem ( ! and ! )

Case 1 ~ With ! ,

!

Therefore, ! converges to ! when ! .

Case 2 ~ With ! ,

!

Therefore, ! converges to ! when ! .

(a + bx)n = nCr ar (bx)n−r , ∀n∈!+

r=0

n

∑= nC0a

n (bx)0 + nC1an−1(bx)1 + nC2a

n−2 (bx)2 + ...+ nCn−1a1(bx)n−1 + nCna

0 (bx)n

(1+ x)n = 1+ nx + n(n −1)2!

x2 + n(n −1)(n − 2)3!

x3 + ..., ∀n∉!+

(a + bx)n = an 1+ bxa

⎛⎝⎜

⎞⎠⎟n

= an 1+ nX + n(n −1)2!

X 2 + n(n −1)(n − 2)3!

X 3 + ...⎛⎝⎜

⎞⎠⎟ , ∀n∉!+ , with X = bx

a

bxa

<1

n = −1 x = ±1

n = −1

(1+ x)−1 = 1− x + x2 − x3 + ...+ (−1)k xk + ..., k ∈{0,∞}

1− x + x2 − x3 + ... 11+ x

−1< x <1

n = −1

(1− x)−1 = 1+ x + x2 + x3 + ...+ xk + ..., k ∈{0,∞}

1+ x + x2 + x3 + ... 11− x

−1< x <1

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B.1 The Binomial Theorem

1. Using binomial expansions, find the linear approximations to the following functions in the immediate neighbourhood of ! .

(a) !

(b) !

(c) !

2. Expand the following functions in ascending powers up to and including the term in ! . In each case, state the range of values of ! for which the expansion is valid.

(a) !

(b) !

B.2 Binomial Coefficients

3. Write down the term indicated in the binomial expansion of each of the following functions.

(a) !

(b) !

(c) !

4. Write down the term indicated in the binomial expansion of each of the following functions.

(a) !

(b) !

(c) !

x = 0

1− 5x( )10

2 − x( )6

1− x( ) 1− x( )10

x3 x

1− 5x( )−2

1− 2x( )−23

1− 4x( )6 , 3rd term

3a + 9b( )8 , 2nd term

a + b( )5 , term in a3

y2− 2x , 3rd term

3a − 9b( )−4 , 2nd term

a + b( )−1 , term in a3

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5. Show that the coefficient of the term in ! of the product ! is 96.

6. For the binomial expansion of ! , show that the ratio of the term in ! to

the term in ! is ! .

Please continue …

x5 x + 2( )5 x − 2( )4

2x + 3( )20 x6

x7 34x

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SET C BINOMIAL SERIES EXPANSIONS

C.1 Cases when !

7. Write down the first 4 terms in the binomial expansion of

(a) !

(b) !

8. Develop the binomial expansion of each of the following function as a series in ascending powers of ! as far as, and including, the term in ! . For each case, establish the necessary conditions for which your result is valid.

(a) !

(b) !

9. ** Using the binomial expansion of ! to show that, correct to 3 d.p.,

! .

10. Determine the first 3 terms in the binomial expansion of ! . Hence, obtain

an estimate for ! .

11. Using the Binomial Theorem, in ascending powers of x, expand and simplify ! . Hence or otherwise, deduce the expansion of ! .

12. Using the Binomial Theorem, find the first 4 terms in descending powers of x of the expression ! . (Compute the coefficients correct to 3 sig. figs.).

n∈!

1+ 3x( )12

2 − 32x⎛

⎝⎜⎞⎠⎟5

x x2

1+ x( ) 1− x( )9

2 + x( ) 1− x2

⎛⎝⎜

⎞⎠⎟20

2 + x( )7

2.08( )7 ≈168.439

1+ 2x( )10

1.0110

(x + 3)4 (2x − 6)4

(0.25x + 3.05)6

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13. Compute the 7th and 9th terms in the expansion of (using

the Binomial Theorem).

14. Given that the Middle Term of is 24 and 2nd Term of is 12,

find possible values of the variables ! and ! . (What single factor determines

any differences in the solution set?)

Please continue …

f (p,q) = 2p − 13q

⎛⎝⎜

⎞⎠⎟

12

2x − 3y( )4 2x + y8

⎛⎝⎜

⎞⎠⎟6

x y

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C.2 Cases when !

15. Expand the following functions in ascending powers up to and including the term in ! . In each case, state the range of values of ! for which the expansion is valid.

(a) !

(b) !

(c) !

16. Find the first 3 terms of the (binomial) expansion of ! . Hence, estimate the

value of ! to 4 d.p.

17. Find the quadratic approximation of ! for ! values close to 0.

18. Provided that ! is small enough to neglect higher powers than 2, then

(a) Show that ! .

(b) Develop the validity condition for your solution.

19. Develop the binomial expansion as a series in ascending powers of ! as far as,

and including, the term in ! for ! . Hence, by letting ! ,

compute an approximation of ! to 4 decimal places.

n∉!+

x3 x

1+ 2x1− 2x

1+ 1x

⎛⎝⎜

⎞⎠⎟−1

3x + 21− 2x4

1+ x( )32

1.02( )32

g(x) = 1

1− 4x( )23x

x

123+ x( ) 1− x( )2

≈ 4 + 203x + 88

9x2

x

x2 1+ x4 + 1− x4 x = 116

174 + 154

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20. Is ! is very small, find the cubic approximation for ! . State the validity

condition for your solution.

21. By substituting 0.08 for ! in ! and its expansion, find ! correct to 4

significant figures.

- ENFIN -

x 110 − x( )4

x 1+ x( )1/2 3

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