Series and Sequences

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An arithmetic series (or progression) is a series where each term is obtained by adding on a constant to the previous term.

The series goes up in equal steps.

The constant is called the common difference (d).

An arithmetic series starts with a first term, a.

The second term is a + d, where d is the common difference.

The third term is a + 2d, and so on.

nth term = (a+(n-1)d)

The sum to n terms is n/2 (2a+(n-1)d) or n x (a+l)/2

A geometric series (or progression) is a series where each term is obtained by multiplying the previous term by a constant number.

The constant is called the common ratio (r).

nth term = a x r^n-1

Sn = a(r^n -1)/r-1

Sum to infinity = a/1-r

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Binomial Expansion

The coefficients are numbers in a line of Pascal's triangle.

The powers of a go down in ones.

The powers of b go up in ones.

The numbers in Pascal's triangle are generated by the formula nCr.

nCr is the number of ways of choosing r objects from a group of n objects.

nCr = n!/r!(n-r)!

The binomial expansion of (a + b)^n is given by:

nC0a^n + nC1a^n-1b + nC2a^n-2b^2 +.....+ nCnb^n