I would recommend always try at least 2 terms, because you could always fluke one!įind the nth term of the quadratic sequence 1, 3, 9, 19, …įirst, find a – the difference of the differences divided by 2. The third term is a × 3 2 + b × 3 + c 9 a + 3 b + c. The second term is a × 2 2 + b × 2 + c 4 a + 2 b + c. The first term is a × 1 2 + b × 1 + c a + b + c. The constants b and/or c might be zero but a definitely isn’t. It’s always a nice feeling, not just in maths, when you give an answer and you know it is correct. A quadratic sequence is a sequence for which the n th term is a n 2 + b n + c. N = 4 4 2 – 2×4 + 4 = 16 – 8 + 4 = 12 4 out of 4! Free resources for teachers and students to hopefully make the teaching and learning of mathematics a wee bit easier and more fun. Let’s do the fourth term as well, we know this should be 12… N = 1 1 2 – 2×1 + 4 = 1 – 2 + 4 = 3 this matches our sequence! This allows us to check the formula we calculated is correct. Likewise, we know that the second term in the sequence is 4, so if we plug 2 into the formula we should get 4. So, if we plug 1 into the formula we should get 3. We know from the question that the first term in the sequence is 3. Going back to why the nth term formula is useful, remember that the formula tells you any term in the sequence. What I would strongly recommend at this stage is that you check your answer. So the nth term of the green sequence is -2n + 4.Īdding this on to what we already knew, this means our nth term formula is n 2 – 2n + 4. The sequence has a difference of -2, and if there were a previous term it would be 4. If you need a reminder of how to find the nth term of a linear sequence, you can re-read the previous blog. We say that the second difference is constant. Consequently, the 'difference between the differences between the sequence's terms is always the same'. We will need to add this on to n 2 – this will tell us our b and c. Quadratic sequences of numbers are characterized by the fact that the difference between terms always changes by the same amount. In other words, a linear sequence results from taking the first differences of a quadratic sequence. This sequence has a constant difference between consecutive terms. It is important to note that the first differences of a quadratic sequence form a sequence. What we now need to do is find the nth term of this green sequence. Any sequence that has a common second difference is a quadratic sequence. This sequence should always be linear – if it isn’t, you have done something wrong. The differences between our sequence and the sequence n 2 now forms a linear sequence (in green above).
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