Examples of quadratic sequences12/30/2023 ![]() ![]() The result equals the middle number squared, then times by $2$. ![]() Rubayat, Hondfa, Jacob and Nathan represented the problem numerically and in words: $(n-1)(n+1)= n^2-1$, so $(n-1)(n+1)+1=n^2$Ĭhloe, Sophia and Shreya from North London Collegiate School made a clear diagram with a good explanation: Peter's proof was different to Charlie's: The pattern is what Charlie quoted,"If you multiply two numbers that differ by 2, and then add one, the answer is always the square of the number between them!" $3 \times 5 + 1= 16$ or $4^2$. Rubayat, Hondfa, Jacob and Nathan from Greenacre Public School in Australia noticed that Maddy and Grace from the Stephen Perse Foundation in the UK continued the numerical pattern: The shaded area in the right hand diagram needs to be 'subtracted' from the green area. "If you choose four consecutive numbers and subtract from the product of the two middle ones the product of the other two, then you always get $2$." Peter from Durham Johnson School in the UK noticed that the expressions all simplify to 2: Tiago and Finn's diagram representation looked like this: Marc and Yang called the numbers $x$ and $y$ and noticed thatĪmrit's proof used Marc and Yang's idea to prove Peter's representation in words. The answer is always the sum of the digits. If you multiply the two consecutive numbers, which differ by one, then add the bigger number to that you're always going to get the square of the bigger number. Viktor and Matija spotted a numerical pattern and summarised it in words: ![]() Try checking it by working out, for example, the 3rd term and checking it with the sequence.Amrit from Hymers College in the UK, Radi and Camilla and Viktor and Matija from European School Varese in Italy, Rubayat, Hondfa, Jacob and Nathan from Greenacre Public School in Australia, Ryan from Dulwich College Seoul in Korea and Peter from Durham Johnson School in the UK sent in good work on this problem. Now that we have found the value of □, we know the □ th term = 2 □ 2 + 1 So, substituting that into the formula for the □ th term will help us to find the value of □: We know that the □ th term = 2 □ 2 + □ □ + 1 Where □ is the 2 nd difference ÷ 2 and □ is the zeroth term We calculated the zeroth term as 1 and the 2 nd difference as 4. So the first difference between the terms in position 0 and 1 will be 6 − 4 = 2. Working backwards, we know the second difference will be 4. The zeroth term is the term which would go before the first term if we followed the pattern back. How do you find the □ th term of a quadratic sequence? We see why it’s called a quadratic sequence the □ th term has an □ 2 in it. The □ th term of a quadratic sequence takes the form of: □ □ 2 + □ □ + □. What is the □ th term of a quadratic sequence? Higher Sequences Digital Revision Bundle What is a quadratic sequence?Ī quadratic sequence is one whose first difference varies but whose second difference is constant. ![]()
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