![]() You may also see closed form sequences and sigma notation during this second year. In an arithmetic series the terms change by a common factor, whereas in a geometric series, they change by a common factor. In addition to binomial expansion with negative/fractional powers, you will study arithmetic and geometric series. Note that the formulae for binomial expansions is given in the formula booklet: Secondly, as mentioned above, negative/fractional powers are studied in Year 2. You have already met arithmetic and geometric series and applied the formulae for their series: We will. A sequence is also referred to as a progression, which is defined as a successive arrangement of numbers in an order according to some specific rules. sequence (the sequence may be finite or infinite). Firstly, positive integer powers of the expansion are studied in Year 1 – this is the only topic we study in Sequences and Series in Year 1. In mathematics, sequence and series are the fundamental concepts of arithmetic. Binomial Expansion is studied in both years of A-Level Maths. Examples of infinite series include binomial expansions when powers of your binomial are negative or fractional or both. Of course, you can have an infinite sequence but for an infinite series to exist, the summation must converge. The difference between a sequence and a series is that the terms in a sequence are listed whereas the terms in series are summed. Recall that during GCSE Maths you were taught the nth term for linear and quadratic sequences and you also looked at compound interest. When we cancel terms in this way we are using what is called the method of differences.You have seen some Sequences and Series before. If we look at the first few terms of this: we see that all of the terms in the middle cancel out, leaving us with just, i.e. Using the partial fractions method, let’s rewrite (do you remember how to do this?) Sequences: A finite sequence stops at the end of the list of numbers like a 1, a 2, a 3, a. If we look at the first few terms, it is hard to spot a pattern. Sequence and Series Formula Types of Sequence and Series. In order to identify the finite sum, we will have to spot patterns within the series. We will now look at infinite series with a finite sum. Each number in the sequence is called a term (or sometimes 'element' or 'member'), read Sequences and Series for a more in-depth discussion. A Sequence is a set of things (usually numbers) that are in order. An explicit formula for this arithmetic sequence is given by an a+(n1)b, n N, a recursive formula is given by a1 a and an an1+ b for n > 1. Standard Summations Reminder (in formula book, but also good to memorise)ġ,2. To find a missing number in a Sequence, first we must have a Rule. Similar to our last example, let’s start with the binomial expansion (r+1) 4=r 4+4r 3+6r 2+4r+1, and substitute in successive natural numbers, giving:įollowing the same procedures as above, we can rearrange this to find We notice that both sides of the equation include so we can subtract this leaving which we can then solve to give a formula for 1 3) to both aides of the equation to give us: If we add all of the terms on the right hand side, we have If we substitute 1,2,3, …,n into this equation we get: Let’s work with the binomial expansion (r+1) 3=r 3+3r 2+3r+1 We can use this to deduce as this is the sum of the first 2n terms minus the first n terms, so is, which isįind an expression for the first n terms of: 1 + 3 + 5 + 7 + … We are not restricted to “n” as our upper limit, for instance would be an arithmetic sequence with first term 1 and difference 1, which from the formula above has a sum of However, is quite different and means the sum of the digits from 1 to n, i.e. Means that k is added together n times, i.e. ![]() We will build on and extend this work, by looking at convergent series and series of squares and cubes of numbers. You have already met arithmetic and geometric series and applied the formulae for their series: A series is the sum of all the terms in a sequence (the sequence may be finite or infinite).
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