If we combine an 2 with bn + c, we get the n th term rule of our quadratic sequence 2 n 2 + 3 n + 1. It gives us the linear sequence 4, 7, 10, 13, 16, which has an n th-term rule of 3 n + 1 (a common difference of 3 and the 0 th term of 1). Now, we can use the value of n and a to find the linear part of this quadratic sequence this will let us find the values of b and c: n We already know n for each term, and a is easy to find: it’s simply the second difference divided by 2. If we remove that, we are left with a linear sequence. The an2 gives the quadratic part of a quadratic sequence in the n th-term rule. We’ll start by finding the n th term rule of the quadratic sequence 6, 15, 28, 45, 66. There are a couple of ways to do this, but in either case, we first must find the first and second differences. To find this rule, we must find a, b and c. The n th-term rule of a quadratic sequence can always be written as an 2 + bn + c. Once you’re left with only additions and subtractions, carry them out in the order they are given:įinding the n th term rule of a quadratic sequence: We can work this out step-by-step, remembering to use the order of operations – indices first, then multiplications, then additions and subtractions: We can substitute n = 9 into the n th term rule: In this case, we are looking for the term 9, so n = 9. So, for example, say we want to find the 9th term in the sequence with nth term rule 3 n 2 – 5 n + 3. So we substitute n into the n th term rule for our quadratic sequence. Remember, n stands for the position of the term: for the 1st term, n = 1 for the 10th term, n = 10 for the 1765th term, n = 1765. In any sequence, if you know the n th term rule, you can find any term in that sequence. It, in turn, means that the next term must be 45 + 14, or 59:įinding any term in a quadratic sequence using an n th term rule: So, for example, if we know the second difference is 2, that means the difference between the last term, 45, and the next term must be 12 + 2: The differences between the differences, or the ‘second differences,’ are always the same in a quadratic sequence this lets us find the next term. The difference between each pair of terms is different, but you may notice the differences themselves form a pattern – they go up by two each time: Say we wanted to see the next term in the quadratic sequence 9, 15, 23, 33, 45: Unlike a linear sequence, the terms in a quadratic sequence do not have a common difference.įinding the following term ( n th term) in a quadratic sequence: A quadratic sequence is a sequence where the n th term rule includes an n 2 (remember, a term is a word for a number in a sequence).
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