Quick binomial expansion using GC

Binomial expansion can be done quickly using the GC.

Consider binomial expansion of Tex2Img_1396931745, where a is a real number, and q is a fraction or negative integer.

The recurrence formula for the coef works out to be

Tex2Img_1396931939

Example 1

Expand Tex2Img_1396932117  in ascending powers up to and including the term in Tex2Img_1396932205

Using TI 84 Plus (OS 2.55)

bin

Therefore, the expansion is

Tex2Img_1396933315

 

The recurrence formula can also be used to expand in descending powers of x.

Example 2

Expand Tex2Img_1397528779 in descending powers of x up to the 7th term.

Using TI 84 Plus (OS 2.55)

1

Therefore, the expansion is

Tex2Img_1397529466

 

Choosing a suitable value for binomial approximation

1

Example 1

2

3

If the expansion is of other forms, can try the method shown in example 2.

Example 2

Suppose you are asked to expand Tex2Img_1396431523 using binomial expansion, then use the results to approximate Tex2Img_1396431669

How do you select a suitable value for the approximation?

Step 1

Make sure the value selected is within the range of values for which the expansion is valid. In this example, the valid range is from -4 to 4.

Step 2

4-x = (13 / perfect square) or (perfect square/13)

So 4-x = (13 /4 or 13/9) or (49/13)

x= 3/13 or 3/4 or 23/9

Choose the smallest value 3/13 as value closer to 0 gives better approximation.

Therefore, Capture

LHS can be approximated using the binomial expansion. Hence Tex2Img_1396431669 can be approximated.

Finding the general term in binomial series

In Amath, finding the general term in binomial expansion is easy, because we can easily evaluate nCr.  However, finding the general term in H2 math can be much more challenging because n can be a fraction or negative integer.

For example

binomial series

 

binomial series solution

 

These type of questions have appeared in JC assignments. Fortunately, it has not appeared in A level exam for the past 10 years. Nevertheless, it is still good to learn how to derive the general term 🙂

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