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Sequences Revision

Sequences Revision. Learning Objective: Arithmetic Sequences Geometric Sequences Nth terms Sums. Arithmetic Sequences. An arithmetic sequence has an recurrence relationship of the form:. u 1 = a number u n+1 = u n + d.

rolando-alban
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Sequences Revision

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  1. Sequences Revision Learning Objective: Arithmetic Sequences Geometric Sequences Nth terms Sums

  2. Arithmetic Sequences An arithmetic sequence has an recurrence relationship of the form: u1 = a number un+1 = un + d The number added on each time is called the common difference.

  3. Arithmetic Sequences Which of these are arithmetic sequences? un+1 = 2un - 4 un+1 = un - 4 un+1 = un + 11 un+1 = 11un - 4 un+1 = (un)2

  4. Geometric Sequences A geometric sequence has an inductive definition of the form: u1 = a un+1 = r un The number multiplied by each time is called the common ratio.

  5. 1st term Common ratio Geometric Sequences The sequence 3, 12, 48, 192, 768 can be defined by… u1 = 3 un+1 = 4 un u2 = 3 x 4 u2 = 12 u3 = 3 x 4 x 4 u3 = 48 u4 = 3 x 4 x 4 x 4 u4 = 192 un = 3 x 4n-1

  6. un = u1 + (n-1)d Sequences For the sequence : 6, 11, 16, ……, 731 How many terms? What sort of question? nth term of arithmetric sequence u1 = 6 d = 5 un = 731 731 = 6 + 5(n-1) n -1 = (731-6)/5 = 145 n = 146

  7. un = u1 x r n-1 Sequences If I put £400 in a bank account on my 16th birthday and get 5% interest per year. How much money would I have on my 41st birthday? What sort of question? nth term of geometric sequence u1 = 400 r = 1.05 n = 25 un = 400 x 1.0524 un = £1290.04

  8. Sn = n/2 (2u1 + (n-1)d) Sequences For the sequence: 7, 9, 11, 13, ….. What is the sum of the first 50 terms? What sort of question? Sum of terms in arithmetic sequence 50 u1 = 7 d = 2 n = S50 = 25(14 + 49 x 2) S50= 25 x 112 = 2800

  9. S10 = u1 (r n – 1)/(r-1) Sequences The sequence: 2, 4a, 8a2, …. Find an expression for the sum of the first 10 terms What sort of question? sum terms of geometric sequence u1 = 2 r = 2a n = 10 S10 = 2((2a)10-1)/(2a-1) S10 = (211a10-2)/(2a-1)

  10. u1 = 12 The 1st term of a geometric sequence is 12 and the sum to Infinity is 9. Find the ratio of terms 12 = 9(1-r) 12 = 9 - 9r 9r = 9 - 12 = -3 r = - 1/3

  11. Converging series A geometric series converges to a limit when.. Means the size of r including negatives

  12. Converging series Which of the following values of r will converge to a limit? 0.9 0.005 -1.5 -0.03 -1 0.3 -0.2

  13. 1/2 Example 3 Evaluate: r = u1 = (1/2)0 = 1 1st term, when n=0

  14. Different sorts of question to try Sum to infinity 1 + 2/5 + 4/25 + … r = 2/5 u1 = 1 sum = 1 / 3/5 = 5/3 (1/3)1=1/3 r = u1 = 1/3 sum = 1/3 / 2/3 = 1/2 As a fraction r = 1/10 u1 = 0.8= 4/5 sum = 4/5 / 9/10 = 8/9

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