Quadratic Sequences Starter: 25/05/2018 n2 + 3n + 2 2n2 + 7n - 5 Find the first four terms for the following sequences when given the position to term rule. 1 n2 + 3n + 2 2 2n2 + 7n - 5 3 3n2 - 2n + 1 6,12,20,30 4,17,34,55 2,9,22,41 4 2n2 - 4n - 10 5 -n2 + 4n 6 -2n2 - 3n + 2 -12,-10,-4,6 3,4,3,0 -3,-12,-25,-42

Quadratic Sequences What… Main Objective: 25/05/2018 Quadratic Sequences Main Objective: What… Find the position to term rule (nth term rule) for quadratic sequences Level 8 Is the answer to: Is the answer to: Is the answer to: Generate the first five term if the nth term rule is:   T(n) = 2n2 + 3n – 4 Find the nth term rule for the following quadratic sequence.   3, 13, 31, 57, 91… Find the nth term rule for the following geometric sequence.   3, 6, 12, 24… Level 7 Level 6 Level EP Coefficient Nth term Position-to-term rule Quadratic Second difference Keywords

Quadratic Sequences Introduction: 25/05/2018 Look at this quadratic sequence: 3 7 13 21 31... +4 +6 +8 +10 The first difference increases: +2 +2 +2 But the second difference remains constant: A sequence is linear if the first difference is constant. A sequence is quadratic if the second difference is constant. The position-to-term rule of a quadratic sequence is always of the form: T(n) = an2 + bn + c Where the coefficients a, b and c are constants and a ≠ 0. The coefficient a is always half the value of the second difference.

Examples: Find the position to term rule for the sequence: Eg1 Find the position to term rule for the sequence: 2, 9, 20, 35, 54... Eg2 Find the position to term rule for the sequence: 4, 1, 0, 1, 4... Consider the first and second difference: Consider the first and second difference: 2 9 20 35 54... 4 1 0 1 4... +7 +11 +15 +19 -3 -1 +1 +3 +4 +4 +4 +2 +2 +2 The second difference is 4, therefore a=4÷2=2 The second difference is 4, therefore a=2÷2=1 T(n) = 2n2 + ... T(n) = n2 + ... To determine the rest of the formula, subtract 2n2 from each term of the sequence: To determine the rest of the formula, subtract 2n2 from each term of the sequence: Sequence: 2 9 20 35 54 Sequence: 4 1 1 4 2n2: 2 8 18 32 50 n2: 1 4 9 16 25 New Seq: 1 2 3 4 New Seq: 3 -3 -9 -15 -21 The position to term rule for the new sequence is: The position to term rule for the new sequence is: T(n) = n - 1 T(n) = -6n + 9 Therefore the overall position to term rule is: Therefore the overall position to term rule is: T(n) = 2n2 + n - 1 T(n) = n2 - 6n + 9

Main Activity: Find the position to term rule for each of these quadratic sequences 1 4, 9, 16, 25... 2 5, 12, 21, 32... 3 2, 18, 40, 68... n2 + 2n + 1 n2 + 4n 3n2 + 7n – 8 4 5, 8, 15, 26... 5 -9, -7, -1, 9... 6 -13, -10, -1, 14 2n2 - 3n + 6 2n2 - 4n - 7 3n2 - 6n – 10 7 0, -2, -6, -12... 8 -2,-4,-10,-20... 9 -10,-21,-36,-55... -n2 + n -2n2 + 4n – 4 -2n2 - 5n – 3 Problem Solving: Jane was in the jungle when she found a strange baby insect. She took it back to her lab and observed its growth over four years. Jane notices at pattern. She decides to work out how many spots the insect will have at the age of 10. How many spots does the insect have after 10 years? 137 spots NRICH: Power Mad Can you find convincing arguments that explain why all the statements below are true? 21 + 31, 23 + 33, 25 + 35, .... 299 + 399 are all multiples of 5 199 + 299 + 399 + 499 is a multiple of 5 1x + 2x + 3x + 4x + 5x is a multiple of 5 when x is odd