Binomial Expansion

(a+b)ⁿ is called a binomial because it has two terms.

Consider the expansions (a+b)¹, (a+b)², (a+b)³, (a+b)⁴ (yes, let’s fully expand them and inspect the result). If you look at the coefficients of these do you notice any pattern?

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Binomial coefficients

The numbers in Pascal’s triangle are actually the binomial coefficients , where n is the row of Pascal’s triangle and k goes from 1 to n from left to right in the triangle.

We can calculate these manually, or using the calculator’s buttons ‘‘ or ‘!’

In general, for all natural numbers n:

It is worth noticing that for all n, , , and for all n and k:

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Arithmetic progression

• We use the following notation:
  • a or a₁ refers to the first term;
  • d refers to the common difference;
  • l referes to the last term.

Note that the common difference may be zero or a negative number.

The nth term is a + (n-1)d.

We use a method developed by Gauss to derive the formula for the sum of the first n terms in an arithmetic sequence. (Let’s do it!)

We should also be aware that the nth term in an arithmetic sequence is S_n-S_n-1

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Geometric Progression

• We use the following notation:
  • a or a₁ refers to the first term;
  • r refers to the common ratio;

The nth term is ar^n-1

We can multiply the whole sum by r and then subtract this to find the formula for the sum of n terms (Let’s do this!)

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Infinite Geometric Progressions

If the absolute value of the common ratio is less than 1, then an infinite geometric series will converge.

S_∞ =

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Combining Arithmetic Series and Geometric Series in a Question

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Below is an extract from the formula book “MF19” that shows all the relevant formulae to this topic that you will be given in the exam:

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