A term formula can be used to generate numbers which form a pattern, or sequence. Here's a simple example: tn = 2n + 5 In this formula, n can be any Natural number, beginning with 1. The first term in this sequence, t1, can be determined by substituting 1 into the formula: t1 = 2(1) + 5 = 2 + 5 = 7 The value of n is the term number. Here 7 is term 1 The next few terms in the sequence can be generated in a similar way: t2 = 2(2) + 5 = 4 + 5 = 9   t3 = 2(3) + 5 = 6 + 5 = 11   t4 = 2(4) + 5 = 8 + 5 = 13 So the sequence formula   tn = 2n + 5   generates the sequence  7, 9, 11, 13, ... For example, term 4 in this sequence is 13. You can use this formula to discover any term in the sequence you want. For example, the 97th term would be: t97 = 2(97) + 5 = 194 + 5 = 199 Let's look at this sequence one more time: 7, 9, 11, 13, .... Notice the obvious pattern that exists from one term to the next ... '... each term is 2 more that the term before it'. This rule is also a formula. It can be used to predict the next term, and the one after that, just by adding 2 (in this example) over and over again. This type of rule is called a recursive formula, because in order to use it to find a term, you must know the previous term. Recursive formulas aren't very useful if you want to work out one specific term in the sequence, because you'll need to know all the terms that came before. For example, consider the sequence  7, 9, 11, 13, ... again. The recursive rule is 'add 2 every time'. What is the 198th term? You have no idea. In order to find out, you will first have to find the first 197 terms! For this reason, recursive formulas aren't very useful. We will be examining only term formulas, which can be used to discover the value of any term you want. There are an infinite number of term formulas, and each one generates a unique sequence. Here's another example: tn = 3 x 2n -1 Let's work out the first few terms of this new sequence: t1 = 3 x 21-1 = 3 x 20 = 3 x 1 = 3 t2 = 3 x 22-1 = 3 x 21 = 3 x 2 = 6    t3 = 3 x 23-1 = 3 x 22 = 3 x 4 = 12    t4 = 3 x 24-1 = 3 x 23 = 3 x 8 = 24 So the sequence formula   tn = 3 x 2n -1 generates the sequence  3, 6, 12, 24, ... The formula for a sequence can be used to discover more terms in the sequence ... as many terms as you want. As a result, whenever a sequence of numbers is listed, quite often the sequence will include the formula embedded within it. For example: 2, 5, 8, 11, ... tn=3n-1 ... You can now use the formula to work out any term you want. For example, the 20th term would be  t20 = 3(20) - 1 = 59 Discovering the term formula for a particular sequence: Here's is another sequence: 28, 45, 62, ... Can you discover the term formula for this sequence, and use it to predict the 50th term? You might notice that 17 is being added every time. To get each term, add 17 to the previous term. But that's the recursive formula, and won't help much to find the 50th term! What's the term formula? The term formula can be found by using this method: We know it will look like this: tn = m·n + b where m is the amount it goes up or down from one term to the next. In this example. m is +17, so we know the term formula is tn = +17n + b  (where b could be positive or negative) We just have to work out b Try working out the first term. t1 =17(1) + b    or  t1 =17 + b Since we know the first term is 28, the value of b must be +11. So the term formula is tn = 17n + 11 tn = 17n + 11 t1 = 17(1) + 11 = 28      t2 = 17(2) + 11 = 45      t3 = 17(3) + 11 = 62 So the 50th term would be 17(50) + 11 = 865 Many things in the real world can be described using sequences. Knowing the sequence formula can help you to solve problems easily. Fortunately, most real problems can be described just two types of sequences, whose formulas always look the same. By learning about these two types of sequences and what their formulas look like, you can easily come up with a term formula for a sequence, and use it to solve real problems. The two types of sequences we will be looking at are called 'arithmetic' and 'geometric' sequences. Go Back