Chapter Contents

• Series and the Binomial Theorem
• 1. Arithmetic Progressions
• 2. Geometric Progressions
• 3. Infinite Geometric Series
• 4. The Binomial Theorem
• Series and the Binomial Theorem Problem Solver

• Comments, Questions?

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From the math blog...

4. The Binomial Theorem

by M. Bourne

A binomial is an algebraic expression containing 2 terms. For example, (x + y) is a binomial.

We sometimes need to expand binomials as follows:

(a + b)⁰ = 1

(a + b)¹ = a + b

(a + b)² = a² + 2ab + b²

(a + b)³ = a³ + 3a²b + 3ab² + b³

(a + b)⁴ = a⁴ + 4a³b + 6a²b² + 4ab³ + b⁴

(a + b)⁵ = a⁵ + 5a⁴b + 10a³b² + 10a²b³ + 5ab⁴ + b⁵

Clearly, doing this by direct multiplication gets quite tedious and can be rather difficult for larger powers or more complicated expressions.

Pascal's Triangle

We note that the coefficients (the numbers in front of each term) follow a pattern. [This was noticed long before Pascal, by the Chinese.]

You can use this pattern to form the coefficients, rather than multiply everything out as we did above.

The Binomial Theorem

We use the binomial theorem to help us expand binomials to any given power without direct multiplication. As we have seen, multiplication can be time-consuming or even not possible in some cases.

Properties of the Binomial Expansion (a + b)ⁿ

• There are n + 1 terms.
• The first term is aⁿ and the final term is bⁿ.

• Progressing from the first term to the last, the exponent of a decreases by 1 from term to term while the exponent of b increases by 1. In addition, the sum of the exponents of a and b in each term is n.

• If the coefficient of each term is multiplied by the exponent of a in that term, and the product is divided by the number of that term, we obtain the coefficient of the next term.

General formula for (a + b)ⁿ

First, we need the following definition:

Definition: n! represents the product of the first n positive integers i.e.

n! = n(n − 1)(n − 2) ... (3)(2)(1)

We say n! as 'n factorial'

The LiveMath expansion of factorials is usually better than our calculators, in that it can go higher. You can go up to about 170!.

LIVEMath

Example 1 - factorial values

Here are some factorial values:

(a) `3! = (3)(2)(1) = 6`

(b) `5! = (5)(4)(3)(2)(1) = 120`

(c) `(4!)/(2!)=((4)(3)(2)(1))/((2)(1))=12`

Note: `(4!)/(2!)` cannot be cancelled down to 2!.

Binomial Theorem Formula

Based on the binomial properties, the binomial theorem states that the following binomial formula is valid for all positive integer values of n:

`(a+b)^n=` `a^n+na^(n-1)b` `+(n(n-1))/(2!)a^(n-2)b^2` `+(n(n-1)(n-2))/(3!)a^(n-3)b^3` `+...+b^n`

This can be written more simply as:

(a + b)ⁿ = ⁿC₀aⁿ + ⁿC₁a^{n − 1}b + ⁿC₂a^{n − 2}b² + ⁿC₃a^{n − 3}b³ + ... + ⁿC_nbⁿ

We can use the `{::}^nC_r` button on our calculator to find these values.

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LiveMath can also find just the coefficients [numbers at the front] for us, too.

These are usually written nCr.

This LiveMath example will find the coefficients nCr. Your calculator can also do the same thing.

LIVEMath

Example 2

Using the binomial theorem, expand (x + 2)⁶.

Answer

Example 3

Using the binomial theorem, expand (2x + 3)⁴

Answer

Example 4

Using the binomial theorem, find the first four terms of the expansion `(2a-1/x)^11`

Answer

Binomial Series

From the binomial formula, if we let a = 1 and b = x, we can also obtain the binomial series which is valid for any real number n if |x| < 1.

`(1+x)^n``=1+nx+(n(n-1))/(2!)x^2``+(n(n-1)(n-2))/(3!)x^3``+...`

NOTE (1): This is an infinite series, where the binomial theorem deals with a finite expansion.

NOTE (2): We cannot use the `{::}^nC_r` button for the binomial series. The `{::}^nC_r` button can only be used with positive integers.

Example 5

Using the binomial series, find the first four terms of the expansion `sqrt(4+x^2.`

Answer