Year 5 Maths Sequences and Series The Key Facts Sequences A sequence is a set of numbers that displays an obvious pattern. By that we mean, it is easy to find the next number in the sequence. The terms in a sequence are often denoted by: u1, u2, u3, ………….ur, …………………….. where ur is the rth term A sequence can be represented in two ways: a) The rule ur = 2r-1 allows us to write down the sequence for all r. b) The iterative rule u1 = 2, ur+1 = ur + 2. The relationship between ur and ur+1 is called a recurrence relation. Convergence, Divergence and Periodic. A sequence is convergent if it’s terms approach one finite value. A sequence is divergent if it is not convergent. A divergent sequence whose terms form a repeating pattern is called a periodic sequence. Series When the terms of a sequence are added, a series is formed: Eg 2 + 4 + 6 + 8 + 10 + …………. is a series. A series can be defined more concisely by using the symbol. Thus 10 r 3 means “the sum of all terms of the form r3, where r takes all integral values from 2 to 10”. r 2 Arithmetic Progressions A sequence where each term exceeds the previous term by a constant amount is called an Arithmetic Progression (or A.P.) The constant amount is called the common difference d. The nth term of an Arithmetic Progression is always given by: a n 1d (where a is the first term) Therefore the fifth term will always be a + 4d, the seventeenth term a + 16d etc. The sum of an Arithmetic Progression The sum of an AP with n terms is given by: Sn = or S n 1 na l where a is the first term and l is the last term 2 1 n2a n 1d where a is the first term and d is the common difference 2 Geometric Progressions A sequence where each term is a common multiple of the previous term is called a Geometric Progression (GP). The constant multiplying factor is called the common ratio. The nth term of a Geometric Progression is given by: ar n 1 where a is the first term and r is the common ratio. The sum of a Geometric Progression The sum of the first n terms of a GP is given by: Sn a 1 rn 1 r where a is the first term, and r is the common ratio.