![]() This is important when finding the term in the sequence given its value as a zero or negative solution for n can be calculated. n represents the term (position) numbers and therefore it can only be positive integers starting from 1 and should also not include 0, n=1, 2, 3, 4, 5, …. $2550 + 2652=5202.$ $51$ squared then times $2$ equals $5202.A common error is to forget to half the second difference before using it as the coefficient of n^ = −1, −4, −9, −16, … has a second difference of −2 but is incorrectly written as 2. For example: $1,2,3.$ $1 \times 2=2,$ $2 \times 3=6$ therefore $6 +2 = 8$. Quadratic sequences can also be called quadratic algebraic sequences. 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: It will reach a maximum vertical height and then fall back to the ground. ![]() 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 $4^2$. Example 3: A ball is thrown upwards from a rooftop, 80 m above the ground. Since there are three unknowns, we need to make. Rubayat, Hondfa, Jacob and Nathan from Greenacre Public School in Australia noticed that Let the nth term, N an2 + bn + c, where a, b and c are constants to be found. Question 13: A sequence has an nth term of n² 6n + 7 Work out which term in the sequence has a value of 23. Maddy and Grace from the Stephen Perse Foundation in the UK continued the numerical pattern: Question 12: A sequence has an nth term of n² + 2n 5 Work out which term in the sequence has a value of 58. 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: ![]() 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. 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. How do you find the th term of a quadratic sequence Look at the sequence: 3, 9, 19, 33, 51, The second difference is 4. 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: 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. ![]()
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