To find the nth term of a sequence

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Presentation transcript:

To find the nth term of a sequence Linear Sequences Aim To find the nth term of a sequence

Mapping Diagrams n 2n 1 2 2 4 6 3 8 4 What do you notice?

Same difference each time Mapping Diagrams n 2n 1 2 2 4 6 3 8 4 Same difference each time Difference is 2

Mapping Diagrams n 4n - 1 1 3 2 7 11 3 15 4 What do you notice?

Mapping Diagrams n 4n - 1 1 3 2 7 11 3 15 4 Same difference difference is 4

Generating a sequence for 2n + 5 An expression can be used to produce a sequence. It is referred to as the nth term TERM 1st term 2nd term 3rd 4th 5th … nth 2n + 5

Generating a sequence for 2n + 5 TERM 1st term 2nd term 3rd 4th 5th … nth 2n + 5 7 (2 x 1 + 5) = 7

Generating a sequence for 2n + 5 TERM 1st term 2nd term 3rd 4th 5th … nth 2n + 5 9 7 (2 x 2 + 5) = 9

Generating a sequence for 2n + 5 TERM 1st term 2nd term 3rd 4th 5th … nth 2n + 5 9 11 7 (2 x 3 + 5) = 11

Generating a sequence for 2n + 5 TERM 1st term 2nd term 3rd 4th 5th … nth 2n + 5 9 11 11 13 7 (2 x 4 + 5) = 13

Generating a sequence for 2n + 5 TERM 1st term 2nd term 3rd 4th 5th … nth 2n + 5 7 9 11 13 15 The sequence generated by 2n + 5 is 7, 9, 11, 13, 15, ……. What do you notice?

Generating a sequence for 2n + 5 TERM 1st term 2nd term 3rd 4th 5th … nth 2n + 5 9 11 11 13 15 7 (2 x 5 + 5) = 15

Generate a sequence for 4n + 3 TERM 1st term 2nd term 3rd 4th 5th … nth 4n + 3

Generate a sequence for 4n + 3 TERM 1st term 2nd term 3rd 4th 5th … nth 7 11 15 19 23 4n + 3

Activity: Complete the sequences TERM 1st 2nd 3rd 4th 5th … nth 6n – 4 5n + 2 2n + 7 n – 4 n + 2

… Solutions 2 6n – 4 7 7 5n + 2 9 2n + 7 -3 n – 4 3 n + 2 1st 2nd 3rd TERM 1st 2nd 3rd 4th 5th … nth 2 8 14 20 26 6n – 4 7 7 12 12 17 22 27 5n + 2 9 11 13 15 17 2n + 7 -3 -2 -1 1 n – 4 3 4 5 6 7 n + 2

What if….. We knew the sequence, how could we find the nth term?

… Finding the nth term 8 How much is each term increasing by? 1st term 3rd 4th 5th … nth 8 10 12 14 16 How much is each term increasing by?

… Finding the nth term 8 + 2 1st term 2nd term 3rd 4th 5th nth 10 12 14 16 + 2

… Finding the nth term 8 The common difference is 2. 1st term 2nd term 3rd 4th 5th … nth 8 10 12 14 16 The common difference is 2. The nth term begins with 2n

… Finding the nth term 8 The sequence generated by 2n is 2 1st term 3rd 4th 5th … nth 8 10 12 14 16 2 4 6 2n

Finding the nth term What would I need to add to the second sequence so that it is equal to the first sequence? TERM 1st term 2nd term 3rd 4th 5th … nth 8 10 12 14 16 2 4 6 2n + ?

… Finding the nth term I would need to add 6 to each term 8 2 + 6 1st term 2nd term 3rd 4th 5th … nth 8 10 12 14 16 2 4 6 2n + 6

… Finding the nth term I would need to add 6 to each term 8 2n + 6 2 1st term 2nd term 3rd 4th 5th … nth 8 10 12 14 16 2n + 6 2 4 6 2n

If….. The nth term is 5n – 1, how could I find the value of the 100th term? Either I could write out all the terms until I reached the 100th term. OR I could substitute n for 100. 100th term is (5 x 100) – 1 = 499