Pattern Sequences and Special Sequences

Worked example

1. 1

   $4,7,10,13$ — Count the sticks in Patterns 1 to 4 and write the counts in order.

2. 2

   $7-4=3$ — Find the gap. Each new square adds 3 sticks, so the difference is 3.

3. 3

   $13+3=16$ — Pattern 5 is Pattern 4 with one more square, so add 3. Pattern 5 has 16 sticks.

4. 4

   $3n$ — For a far pattern, use a rule. 3 sticks per square gives 3n, which is 3, 6, 9, 12.

5. 5

   $4-3=1$ — Compare the first terms: 4 sticks, but 3n gives 3. Every count is 1 more than 3n.

6. 6

   $3n+1$ — So the rule is 3n + 1. The extra 1 is the starting stick on the left.

7. 7

   $3\times 10+1=31$ — Put n = 10 into the rule. Pattern 10 has 31 sticks.

Worked example

1. 1

   $3n+1=64$ — Which pattern has 64 sticks? Set the rule equal to 64.

2. 2

   $3n=63$ — Take 1 from both sides, so the n part is alone.

3. 3

   $n=21$ — Divide both sides by 3. 21 is a whole number, so Pattern 21 has 64 sticks.

4. 4

   $3n+1=50$ — Now test 50 sticks the same way.

5. 5

   $3n=49$ — Take 1 from both sides.

6. 6

   $n=16.33\dots$ — Divide by 3. n is not a whole number, and there is no Pattern 16⅓, so no pattern has 50 sticks.

Worked example

1. 1

   $5,9,13,17,\dots$ — The sequence starts at 5, an odd number, and adds 4 each time.

2. 2

   $\text{odd}+\text{even}=\text{odd}$ — 4 is even, and odd + even is always odd. So every term is odd.

3. 3

   $238\text{is even}$ — 238 ends in 8, so it is even. An even number cannot be a term, so 238 is not in the sequence.

Worked example

1. 1

   $2$ — The first term, given in the question. The rule is × 3, then − 2.

2. 2

   $2\times 3=6$ — First step: multiply by 3.

3. 3

   $6-2=4$ — Second step: subtract 2. So the second term is 4.

4. 4

   $4\times 3-2=10$ — Use both steps again on the new term: 12 − 2 = 10.

5. 5

   $10\times 3-2=28$ — And again: 30 − 2 = 28. The sequence is 2, 4, 10, 28.

Worked example

1. 1

   $82+2=84$ — Undo the last step first: the rule subtracted 2, so add 2.

2. 2

   $84÷3=28$ — Then undo the multiply: divide by 3. The 4th term is 28.

Worked example

1. 1

   $\text{circle, square, triangle, circle, }\dots$ — The block that repeats is 3 shapes long.

2. 2

   $3,6,9,\dots$ — A triangle ends every block, so the triangles sit at the multiples of 3.

3. 3

   $50=3\times 16+2$ — What is at position 50? Divide 50 by 3: 16 full blocks, remainder 2.

4. 4

   $\text{remainder }2\Rightarrow \text{square}$ — After 16 full blocks, count 2 more shapes: circle, square. Position 50 is a square.

Worked example

1. 1

   $2,5,10,17$ — The gaps are 3, 5, 7. They grow, so the sequence is not linear.

2. 2

   $1,4,9,16$ — Write the square numbers underneath to compare.

3. 3

   $2-1,5-4,10-9,17-16$ — Take each square away from the term above it. The answer is 1 every time.

4. 4

   $n^2+1$ — So each term is a square number plus 1: the nth term is n² + 1.

Worked example

1. 1

   $\text{grey: }3,5,7$ — The grey tiles go up by 2 each time, so they are linear.

2. 2

   $2n+1$ — 2 each time gives 2n, which is 2, 4, 6. Each grey count is 1 more, so grey = 2n + 1.

3. 3

   $\text{white: }1,4,9$ — The white tiles are the square numbers, so white = n².

4. 4

   $2n+1+n^2$ — Every pattern is grey plus white, so add the two rules to get the total.

5. 5

   $n^2+2n+1$ — Tidy it up, biggest power first. Check n = 1: 1 + 2 + 1 = 4.

6. 6

   $10^2+2\times 10+1=121$ — Pattern 10 has 121 tiles. Check: 21 grey + 100 white = 121.

Worked example

1. 1

   $S:7,13,19,25,31,37,43$ — Sequence S adds 6 each time. Write enough terms to give a match a chance.

2. 2

   $T=n^2+1:2,5,10,17,26,37$ — For sequence T, put n = 1, 2, 3 … into the rule n² + 1.

3. 3

   $37$ — 37 is in both lists, so 37 is a number in both sequences.

4. 4

   $6\times 6+1=37$ — Check with S's nth term, 6n + 1: n = 6 gives 37, a whole n, so it really is a term.