Pattern Recognition

Show your work for each problem.

1.
What is the next number in the pattern $2, 5, 8, 11, \ldots$ ?
1. a)14
2. b)16
3. c)13
4. d)15
Answer: 14
Each term is $3$ more than the one before: $5 - 2 = 3$ , $8 - 5 = 3$ , $11 - 8 = 3$ . So the next term is $11 + 3 = 14$ .
2.
Which rule fits the pattern $4, 8, 16, 32, \ldots$ ?
1. a)Add 4
2. b)Add 8
3. c)Multiply by 4
4. d)Multiply by 2
Answer: Multiply by 2
$4 \times 2 = 8$ , $8 \times 2 = 16$ , $16 \times 2 = 32$ . Add $4$ works only once: $8 + 4 = 12$ , not $16$ .
3.
Beads are threaded in the order red, blue, green, red, blue, green, and so on. What color is the 14th bead?
1. a)Blue
2. b)Red
3. c)Yellow
4. d)Green
Answer: Blue
The block red, blue, green has $3$ beads. $14 \div 3 = 4$ remainder $2$ , so the 14th bead is the 2nd color of the block: blue.
4.
Which list follows the rule "start at 3, multiply by 3"?
1. a)3, 9, 27, 81
2. b)3, 6, 9, 12
3. c)3, 6, 12, 24
4. d)3, 9, 15, 21
Answer: 3, 9, 27, 81
$3 \times 3 = 9$ , $9 \times 3 = 27$ , $27 \times 3 = 81$ . The list $3, 6, 9, 12$ adds $3$ , $3, 6, 12, 24$ multiplies by $2$ and $3, 9, 15, 21$ adds $6$ .
5.
Which rule fits ALL the terms of $1, 2, 4, 7, 11$ ?
1. a)Add 1 each time
2. b)Add 3 each time
3. c)Multiply by 2
4. d)Add 1, then 2, then 3, then 4
Answer: Add 1, then 2, then 3, then 4
The differences are $1, 2, 3, 4$ : $1 + 1 = 2$ , $2 + 2 = 4$ , $4 + 3 = 7$ , $7 + 4 = 11$ . Multiply by $2$ fails at the 4th term: $4 \times 2 = 8$ , not $7$ .
6.
What is the next number? $40, 35, 30, 25, \ldots$
Answer: 20
This is a subtracting pattern: each term is $5$ less than the one before. $25 - 5 = 20$ .
7.
What is the next number? $1, 3, 9, 27, \ldots$
Answer: 81
Each term is multiplied by $3$ : $1 \times 3 = 3$ , $3 \times 3 = 9$ , $9 \times 3 = 27$ . Next: $27 \times 3 = 81$ .
8.
Find the missing number: $6, 11, \_\_, 21, 26$
Answer: 16
$11 - 6 = 5$ and $26 - 21 = 5$ , so the step is $5$ . The missing term is $11 + 5 = 16$ . Check: $16 + 5 = 21$ ✓.
9.
What is the next number? $2, 7, 4, 9, 6, 11, \ldots$
Answer: 8
The changes are $+5, -3, +5, -3, +5$ . The last change was $+5$ , so next comes $-3$ : $11 - 3 = 8$ .
10.
What is the 10th term of the pattern $7, 12, 17, 22, \ldots$ ?
Answer: 52
The step is $5$ . From the 1st to the 10th term there are $9$ jumps: $7 + 9 \times 5 = 7 + 45 = 52$ .
11.
Find the next two terms: $3, 7, 11, 15, \ldots$
Answer: 19, 23
1. What is the difference between neighboring terms? 4
2. What is the 5th term? 19
3. What is the 6th term? 23
12.
Find the rule and the next term: $5, 10, 20, 40, \ldots$
Answer: 80
1. What is $10 - 5$ ? 5
2. What is $20 - 10$ ? (Notice it is not the same difference) 10
3. Each term is multiplied by which number? 2
4. What is the next term? 80
13.
Find the next term: $2, 3, 5, 8, 12, \ldots$
Answer: 17
1. What is the difference between the 1st and 2nd terms? 1
2. What is the difference between the 4th and 5th terms? 4
3. The differences are 1, 2, 3, 4. What is the next difference? 5
4. What is the next term? 17
14.
Find the next two terms: $20, 24, 22, 26, 24, \ldots$
Answer: 28, 26
1. How much is added from $20$ to $24$ ? 4
2. How much is subtracted from $24$ to $22$ ? 2
3. The last change ( $26$ to $24$ ) was $-2$ . What is the 6th term? 28
4. What is the 7th term? 26
15.
The pattern star, moon, sun, star, moon, sun, ... continues. Which shape is the 25th? Give its position in the block (1 = star, 2 = moon, 3 = sun).
Answer: 1
1. How many shapes are in one block? 3
2. How many full blocks fit into 25 shapes? 8
3. What is the remainder? (This is the position in the block) 1
16.
Figure 1 of a tile pattern has $6$ tiles, Figure 2 has $10$ and Figure 3 has $14$ . How many tiles does Figure 10 have?
Answer: 42
1. How many tiles does each new figure add? 4
2. How many jumps are there from Figure 1 to Figure 10? 9
3. How many tiles do all the jumps add together? 36
4. How many tiles does Figure 10 have? 42
17.
Sam has 12 dollars and saves 3 dollars every week. After how many weeks will he have 48 dollars?
Answer: 12
1. By how much does his money grow each week? 3
2. How much more money does he still need? 36
3. How many weeks of saving make up this amount? 12
18.
Mia says the rule for $2, 6, 18, 54$ is "add 4". Check her rule and find the correct next term.
Answer: 162
1. What is $2 + 4$ ? (Does it match the 2nd term?) 6
2. What is $6 + 4$ ? (The 3rd term is 18, so the rule fails) 10
3. Try multiplying. Each term is multiplied by which number? 3
4. The rule ×3 fits all terms ( $18 \times 3 = 54$ ). What is the next term? 162