Sequences and series

• An arithmetic sequence is one where there is a common difference between each term.
• Arithmetic sequences are in the form a, a + d, a + 2d ... where a is the first term and d is the common difference.
• The sum of the first n terms of an arithmetic sequence is given by Sn = n/2 (2a + (n -1) d) = n/2 (a + l), where a is the first term, d is the common difference and l is the last term.
• The sum of the first n terms in an arithmetic sequence formula can be proved by adding the sum written out normally with the sum written out in reverse, and then using algebra to get the result equal to Sn.
• A geometric sequences is one where there is a common ratio between the terms.
• A geometric sequence is in the form a, ar, ar^2 ... where a is the first term and r is the common ratio.
• The nth term of a geometric sequence is given by Un = a r ^(n - 1).
• The nth term of an arithmetic sequence is given by Un = a + (n - 1) d.
• The sum of the first n terms in a geometric sequence is given by Sn = a (1 - r^n)/ (1 - r).
• The sum of the first n terms in a geometric sequence equation can be proved by subtracting the sequence multiplied by the common ratio from the sequence, and then using algebra to get the result equal to Sn.
• A geometric sequence is convergent if the common ratio is less than one.
• The sum to infinity of a geometric sequence only exists for convergent sequences, and is given by Si = a/(1 - r).