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This really depends on what types of sequence do you get. In GCSE, there are a few sequences you will be frequently tested. These sequences are square number sequence, cube number sequence, Fibonacci sequence, linear sequence(same as arithmetic sequence), quadratic sequence, and geometric sequence(depends on exam boards) Let’s start with square number sequence and cube number sequence: recursive formula for square number sequence is n^2 and the recursive formula for cube number sequence is n^3. Now let’s look at Fibonacci sequence: the recursive is a(n)=a(n-2)+a(n-1), i.e the next number appears in the sequence is the sum of two previous number. Linear sequence/arithmetic sequence and quadratic sequence are the two most common sequence you will be tested in your GCSE exam. The recursive formula for linear sequence is a(n)=a(1)+d(n-1), a(1) means the first term, d means difference between two terms. The recursive formula for quadratic sequence is harder to find. Firstly, the general formula for quadratic sequence is ax^2+bx+c. You need to find a, b, c. In quadratic sequence, the difference of the difference between two terms is constant. For example, a sequence 2, 9, 18, 29, ...... The difference between the 1st and the 2nd term is 7, the difference between the 2nd and the 3rd term is 9, the difference between 3rd and the 4th term is 11, and the difference of the difference is always 2, 7+2=9, 9+2=11, etc. So you need to use the difference of the difference to divide by 2 to work out a. In this case, a=2/2=1. Now substitute n with 1 and 2 to make a simultaneous equation. So in this case two equations will be 1+b+c=2, 4+2b+c=7. Then solve this simultaneous equations to work out b and c, and you have the recursive formula for a quadratic sequence. The last common sequence that might appear in GCSE is geometric sequence, and the recursive formula is a(n)=a(1)*r^(n-1). I hope you find this answer is helpful!!!!!
In the case of an arithmetic sequence (where the difference between each number is constant), the expression used for the “nth term” (or general formula for the sequence) is always: dn + zero term Where “d” is the difference between each term And the “zero term” is the first term minus the difference (what term number zero would be). So for example 3,5,7,9.... = 2n+1 1,2,3,4,5..... =n This general formula you find can then be used to find, say, the hundredth term in a sequence even though you were only given a few terms at first.
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