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Understanding Sequences and nth Term Formulas in Mathematics

This guide explores various numerical sequences and the rules governing their nth terms. It covers sequences based on addition, subtraction, multiplication, and more complex patterns. Specific sequence examples illustrate how to derive the nth term and predict future terms. Practical applications, such as taxi fare calculations, are also discussed to show real-world uses of these mathematical concepts. Engage with exercises designed to reinforce understanding, build problem-solving skills, and enhance appreciation for mathematical sequences.

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Understanding Sequences and nth Term Formulas in Mathematics

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  1. A few sequences… 9, 13, 17, 21…. ….. 25, 29 term to term rule: add 4

  2. A few sequences… 20, 15, 10, 5…. ….. 0, -5 term to term rule: minus 5

  3. A few sequences… 1, 10, 100, 1000…. ….. 10,000, 100,000 term to term rule: x 10

  4. A few sequences… 88, 44, 22, 11…. ….. 5.5, 2.75 term to term rule: half

  5. Sequences the nth term

  6. 1st 2nd 3rd 4th 5th 6th 7th 10, 20, 30, 40, 50, 60, 70…… The position to term rule is: whichever term I’m interested in X 10

  7. 1st 2nd 3rd 4th 5th 6th 7th 4, 8, 12, 16, 20, 24, 28…… The position to term rule is: n whichever term I’m interested in X 4 nth term = n x 4

  8. What is the position to term rule: 2, 4, 6, 8, 10 …. nth term = n x 2 = 2n 6, 12, 18, 24 …. nth term = 6n 5, 10, 15, 20, 25…. nth term = 5n 100, 200, 300, 400…. nth term = 100n What’s the 7th term? 700 What’s the 10th term? 1000 What’s the 18th term? 1,800

  9. more complicated…. 5, 8, 11, 14, 17, 20 ….. +3 +3 +3 +3 +3 common difference is 3 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 + 2 nth term = 3n + 2

  10. To work out the rule for the nth term of a sequence 6, 11, 16, 21, 26… Step 1: Common difference? Step 2: How has the table been shifted? 1 1 2 2 3 3 4 4 5 5 6 6 7 7 8 8 9 9 10 10 11 11 12 12 13 13 14 14 15 15 16 16 17 17 + 1 nth term = 5n + 1

  11. Questions to try: • Work out the rule for the nth term then work out the 100th term • 3, 5, 7, 9, 11, 13…. • 12, 20, 28, 36, 44…. • 19, 29, 39, 49, 59…. • 7, 10, 13, 16, 19…. • 14, 20, 26, 32, 38…. • 55, 60, 65, 70, 75… • 8, 17, 26, 35, 44…. • Extension: • 1, 9, 17, 25, 33…. • -2, 8, 18, 28, 38…. • -2, -4, -6, -8, -10… • 1, 4, 9, 16, 25…. • 3, 6, 11, 18, 27…. !! !!

  12. Real Life Example: • You own a taxi company that charges as follows: • £3.50 for calling the cab • 20p for every minute of journey time Work out a formula for the cost of a journey that’s nminutes long Use your formula to cost a journey of 2 hours

  13. What pattern of matchsticks would follow this sequence rule: 4n + 2

  14. Sequences the nth term

  15. Extension work T(n) = n2 T(n) = 3n2 + n T(n) = 4n2 + n – 1 • For each of these sequences work out the first five terms • What is the first difference? • What is the second difference? • Is there a way of predicting the second difference?

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