26112020 Starter Quadratic Sequences Find the first four

26/11/2020 Starter: Quadratic Sequences Find the first four terms for the following sequences when given the position to term rule. 1 n 2 + 3 n + 2 2 6, 12, 20, 30 4 2 n 2 - 4 n - 10 -12, -10, -4, 6 2 n 2 + 7 n - 5 3 4, 17, 34, 55 5 -n 2 + 4 n 3, 4, 3, 0 3 n 2 - 2 n + 1 2, 9, 22, 41 6 -2 n 2 - 3 n + 2 -3, -12, -25, -42

Quadratic Sequences Main Objective: 26/11/2020 Find the position to term rule (nth term rule) for quadratic sequences Level 8 What… Is the answer to: Generate the first five term if the nth term rule is: T(n) = 2 n 2 + 3 n – 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 Keywords Is the answer to: Coefficient Nth term Position-to-term rule Quadratic Second difference Level EP

26/11/2020 Quadratic Sequences Introduction: Look at this quadratic sequence: 3 The first difference increases: But the second difference remains constant: 7 +4 13 +6 +2 21 +8 +2 31. . . +10 +2 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) = an 2 + 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: Eg 1 Find the position to term rule for the sequence: 2, 9, 20, 35, 54. . . Consider the first and second difference: 2 9 20 35 Eg 2 Find the position to term rule for the sequence: 4, 1, 0, 1, 4. . . Consider the first and second difference: 54. . . The second difference is 4, therefore a=4÷ 2=2 4 1 0 1 4. . . The second difference is 4, therefore a=2÷ 2=1 T(n) = 2 n 2 +. . . T(n) = n 2 +. . . To determine the rest of the formula, subtract 2 n 2 from each term of the sequence: Sequence: 2 9 20 35 54 Sequence: 4 1 0 1 4 2 n 2: 2 8 18 32 50 n 2 : 1 4 9 16 25 New Seq: 0 1 2 3 4 New Seq: 3 -3 -9 -15 -21 The position to term rule for the new sequence is: Therefore the overall position to term rule is: T(n) = n - 1 T(n) = 2 n 2 + n - 1 The position to term rule for the new sequence is: Therefore the overall position to term rule is: T(n) = -6 n + 9 T(n) = n 2 - 6 n + 9

Main Activity: 1 4, 9, 16, 25. . . Find the position to term rule for each of these quadratic sequences 2 n 2 + 2 n + 1 4 5, 8, 15, 26. . . 0, -2, -6, -12. . . NRICH: Power Mad Problem Solving: -n 2 + n 3 n 2 + 4 n 5 2 n 2 - 3 n + 6 7 5, 12, 21, 32. . . -9, -7, -1, 9. . . 3 n 2 + 7 n – 8 6 2 n 2 - 4 n - 7 8 -2, -4, -10, -20. . . -2 n 2 + 4 n – 4 2, 18, 40, 68. . . -13, -10, -1, 14 3 n 2 - 6 n – 10 9 -10, -21, -36, -55. . . -2 n 2 - 5 n – 3 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. 137 spots 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? Can you find convincing arguments that explain why all the statements below are true? a) 21 + 31, 23 + 33, 25 + 35, . . 299 + 399 are all multiples of 5 b) 199 + 299 + 399 + 499 is a multiple of 5 c) 1 x + 2 x + 3 x + 4 x + 5 x is a multiple of 5 when x is odd