Special Sequences

Compare each term with its position in the sequence: 1st, 2nd, 3rd, …

A cube number is a whole number multiplied by itself and then by itself again.

Look at how many dots are added to make each new pattern.

In a Fibonacci sequence, each term is found from the two terms just before it.

Work out what each term is multiplied by to get the next term.

Is each term a fixed amount less than the term before, or a fraction of it?

1. Each term is half of the term before: 12\div2=6, 6\div2=3, 3\div2=1.5
2. The next term is 1.5\div2=0.75

Use the two terms just before the box.

1. The missing term is the sum of the two terms before it: 2+5=7
2. Check: 5+7=12 and 7+12=19

Write down the differences between the terms of each sequence.

1. 3, 7, 11, 15, 19: the differences are 4, 4, 4, 4. They stay the same.
2. 4, 7, 12, 19, 28: the differences are 3, 5, 7, 9. They go up by 2 each time.
3. 2, 4, 8, 16, 32: the differences are 2, 4, 8, 16. They double.
4. 1, 1, 2, 3, 5: the differences are 0, 1, 1, 2.
5. So the answer is 4, 7, 12, 19, 28

Find the differences between the terms, then look at how the differences change.

1. The differences are 2, 4, 6, 8. They go up by 2 each time.
2. The next difference is 10.
3. The next term is 21+10=31

Divide a term by the term before it.

1. 12\div3=4, 48\div12=4 and 192\div48=4
2. Each term is multiplied by 4

Each pattern is a square. How many tiles are along each side?

Part (a)

1. Pattern 1 has 1\times1=1 tile, pattern 2 has 2\times2=4 tiles and pattern 3 has 3\times3=9 tiles.
2. Pattern number 7 is a square 7 tiles by 7 tiles: 7\times7=49 tiles.

Part (b)

1. Pattern number 8 has 8\times8=64 tiles.
2. Pattern number 6 has 6\times6=36 tiles.
3. 64-36=28 more tiles.

The 4th term is the 2nd term plus the 3rd term, so start by finding the 2nd term.

1. 11+\text{2nd term}=18, so the 2nd term is 18-11=7
2. \text{1st term}+7=11, so the 1st term is 11-7=4
3. Check: 4+7=11, 7+11=18, 11+18=29

Find the differences between the terms and how much they go up by.

Part (a)

1. The differences are 4 and 6. They go up by 2.
2. The next difference is 8, so the 4th term is 12+8=20

Part (b)

1. The next differences are 10 and 12.
2. 5th term: 20+10=30. 6th term: 30+12=42

Use the two terms that are next to each other to find the number you multiply by.

1. 375\div75=5, so each term is multiplied by 5
2. The missing term is 3\times5=15
3. Check: 15\times5=75

Add the two terms before to get the next term, collecting the a terms each time.

1. 3rd term: 2a+a=3a
2. 4th term: a+3a=4a
3. 5th term: 3a+4a=7a

Write the 3rd, 4th and 5th terms in terms of a, then make an equation.

1. 3rd term: 3+a
2. 4th term: a+(3+a)=2a+3
3. 5th term: (3+a)+(2a+3)=3a+6
4. 3a+6=36, so 3a=30 and a=10
5. Check: 3, 10, 13, 23, 36

Write the 3rd, 4th and 5th terms in terms of a and add them up.

1. 3rd term: a+4a=5a
2. 4th term: 4a+5a=9a
3. 5th term: 5a+9a=14a
4. Their sum is 5a+9a+14a=28a
5. 28a=168, so a=168\div28=6
6. Check: the sequence is 6, 24, 30, 54, 84 and 30+54+84=168

Keep multiplying by 3 and count the levels as you go.

1. Level 1: 5, level 2: 15, level 3: 45, level 4: 135, level 5: 405, level 6: 1215
2. Level 5 gives less than 1000 points and level 6 gives more.
3. The first level that gives more than 1000 points is level 6

Work out the first difference and the last difference. The four differences go up evenly from one to the other.

1. The first difference is 6-3=3 and the last difference is 27-18=9
2. There are four differences, going up by the same amount each time from 3 to 9: 3, 5, 7, 9
3. So the missing term is 6+5=11
4. Check: 11+7=18

Work out how many dots are in pattern number 10 of each sequence separately.

1. Sequence A is the square numbers 1, 4, 9, \ldots Pattern number 10 has 10\times10=100 dots.
2. Sequence B is the triangle numbers. Each pattern adds a row with one more dot than the row before:
3. 1, 3, 6, 10, 15, 21, 28, 36, 45, 55, so pattern number 10 has 55 dots.
4. 100-55=45 more dots.