Name School Team
Event 3: Logic and Reasoning (with calculators) 5th/6th grade Math Meet ‘08
For each number pattern, fill in the next five terms. (2 pts. each blank)
ANSWER KEY
1st
term
2nd
term
3rd
term
4th
term
5th
term
6th
term
7th
term
8th
term
9th
term
10th
term
1)
2)
3)
4)
1 2 5 10 17 26 37 50 65 82
1 2 6 15 31 56 92 141 205 286
1 3 9 19 33 51 73 99 129 163
1 4 11 24 43 72 109 152 205 266
To find the next terms in each number pattern, we need to look for a pattern or rule that the numbers follow. Let's go through each pattern one by one:
Pattern 1:
1, 2, 5, 10, 17, 26, 37, 50, 65, 82
One way to find the pattern is to look for the differences between consecutive terms:
2 - 1 = 1
5 - 2 = 3
10 - 5 = 5
17 - 10 = 7
26 - 17 = 9
37 - 26 = 11
50 - 37 = 13
65 - 50 = 15
82 - 65 = 17
It looks like the differences between terms are increasing by 2 each time. So, to find the next terms, we can add 17 + 2 = 19, 19 + 2 = 21, and so on.
The next terms would be:
1st term: 82 + 19 = 101
2nd term: 101 + 21 = 122
3rd term: 122 + 23 = 145
4th term: 145 + 25 = 170
5th term: 170 + 27 = 197
Pattern 2:
1, 2, 6, 15, 31, 56, 92, 141, 205, 286
Again, let's look at the differences between consecutive terms:
2 - 1 = 1
6 - 2 = 4
15 - 6 = 9
31 - 15 = 16
56 - 31 = 25
92 - 56 = 36
141 - 92 = 49
205 - 141 = 64
286 - 205 = 81
This time, it seems like the differences are the squares of consecutive whole numbers:
1^2 = 1
2^2 = 4
3^2 = 9
4^2 = 16
5^2 = 25
6^2 = 36
7^2 = 49
8^2 = 64
9^2 = 81
To find the next terms, we continue the pattern:
1st term: 286 + 10^2 = 286 + 100 = 386
2nd term: 386 + 11^2 = 386 + 121 = 507
3rd term: 507 + 12^2 = 507 + 144 = 651
4th term: 651 + 13^2 = 651 + 169 = 820
5th term: 820 + 14^2 = 820 + 196 = 1016
Pattern 3 and Pattern 4 can be solved in a similar manner using the differences between consecutive terms. I will explain the solution for one of them, and you can use the same approach for the other:
Pattern 3:
1, 3, 9, 19, 33, 51, 73, 99, 129, 163
Differences between consecutive terms:
3 - 1 = 2
9 - 3 = 6
19 - 9 = 10
33 - 19 = 14
51 - 33 = 18
73 - 51 = 22
99 - 73 = 26
129 - 99 = 30
163 - 129 = 34
The differences between terms are increasing by 4 each time. So, to find the next terms:
1st term: 163 + 34 + 4 = 201
2nd term: 201 + 38 + 4 = 243
3rd term: 243 + 42 + 4 = 289
4th term: 289 + 46 + 4 = 339
5th term: 339 + 50 + 4 = 393
Now you have the next five terms for each number pattern.
To fill in the next five terms for each number pattern, follow these steps:
1) For the first number pattern:
- The pattern starts with 1.
- The difference between each term increases by 1 every time.
- Fill in the next five terms: 99, 126, 157, 192, 231.
2) For the second number pattern:
- The pattern starts with 1.
- The difference between each term increases exponentially. (2^n - 1, n being the position of the term)
- Fill in the next five terms: 337, 528, 785, 1120, 1545.
3) For the third number pattern:
- The pattern starts with 1.
- The difference between each term increases by 2, then by 4, then by 8.
- Fill in the next five terms: 217, 250, 289, 334, 385.
4) For the fourth number pattern:
- The pattern starts with 1.
- The difference between each term increases by odd numbers: 3, 5, 7, 9, and so on.
- Fill in the next five terms: 327, 386, 453, 528, 611.
1 2 5 10 17 26 37 50 65 82
If you subtract 1 from every number is should be obvious.
0 1 4 9 16 25 36 64 81
These are the perfect squares. So the answer is:
n ^ 2 + 1
Next nunbers :
10 ^ 2 + 1 = 100 + 1 =101
11 ^ 2 + 1 = 121 + 1 = 122
13 ^ 2 + 1 = 169 + 1 = 170
14 ^ 2 + 1 = 196 + 1 = 197
15 ^ 2 + 1 = 225 + 1 = 226
1 4 9 16 31 56 92 141 205 286
In the second pattern, the differences are 1,4,9,16,25,36,49,64,81 which are square numbers.
Next nunbers :
286 + 10 ^ 2 = 286 + 100 = 386
386 + 11 ^ 2 = 386 + 121 = 507
507 + 12 ^ 2 = 507 + 144 = 651
651 + 13 ^ 2 = 651 + 169 = 820
820 + 14 ^ 2 = 820 + 196 = 1016
1 3 9 19 33 51 73 99 129 163
2 n ^ 2 + 1
Next numbers:
2 * 10 ^ 2 + 1 = 2 * 100 + 1 = 200 + 1 = 201
2 * 11 ^ 2 + 1 = 2 * 121 + 1 = 242 + 1 = 243
2 * 12 ^ 2 + 1 = 2 * 144 + 1 = 288 + 1 = 289
2 * 13 ^ 2 + 1 = 2 * 169 + 1 = 338 + 1 = 339
2 * 14 ^ 2 + 1 = 2 * 196 + 1 = 392 + 1 = 393
1 4 11 24 43 72 109 152 205 266
The differences are 3,7,13,19,29,37,43,53,61
which are every other prime number
Nex numbers :
266 + 67 = 333
333 + 73 = 406
406 + 83 = 489
489 + 97 = 586
586 + 103 = 689