 A geometric sequence is one where each term can be found by multiplying by the same number over and over again. The number could be a fraction, and could be positive or negative. Here's an example: The first term in a sequence is called t1, or more commonly a. In the example above, a = 3 The number that is being multiplied each time is 2. This is called r, the common ratio. You can always find a sequence's common ratio by taking any term and dividing by the term directly before it. For example, 24 ÷ 12 = 2. It's because you divide that r is called the common ratio. For any geometric sequence, the term formula always looks like this:  So for the sequence above, where a = 3 and r = 2, As long as the sequence is geometric, and you know a and r, you can always get the term formula for that sequence by filling a and r into tn = a·rn-1 Let's do another example: It's a geometric sequence because to get from one term to the next you always multiply by 1/2 or 0.5 So we have a = 64, r = 0.5, and the term formula will be tn = arn-1 The actual term formula for this sequence is: tn = 64x(0.5)n-1 So the 10th term can now be found: t10 = 64x(0.5)10 = 0.0625 You can also use the geometric term formula backwards, to discover how many terms are in a particular sequence. For example:  There were 2 rabbits on an island during the first year. In the second year there were 6 rabbits. In the third year there were 18 rabbits. When you visit, there are 486 rabbits. How many years have the rabbits been breeding? Always begin a sequence question by writing the sequence of numbers, and then stating what you have to find. Here is the problem in simpler form: How many terms? To get from one term to the next, you must always multiply by 3. So this sequence is geometric. a = 2 r = 3 tn = arn-1 If we designate the last term 486 as tn, then n will be the number of that term. Solving for n will tell us how many terms are in the sequence up to 486. tn = arn-1 486 = 2 x 3n-1 243 = 3n-1 since 243 = 35, n-1 must equal 5. So n = 6. So the sequence 2, 6, 18, ... 486 has 6 terms (This is the same as saying that 486 is the 6th term) The rabbits have been breeding for 6 years. |