Find the missing numbers to complete the patterns. Some are arithmetic and some are geometric. State the commom differences or common ratio.
1) 1,2,___,___,___,32
2)800,80,8,0,8,0.08,___,____
Some tricky ones that don't follow the "rules."
1)17,15,25,23,33,31,_____,____.
2)3,6,7,14,15,30,31,____,____.
3)1,6,5,10,9,14,13,___,___.
Please Help! I was not in class for this! A.S.A.P.! thank you!
The first is not that obvious unless you are familiar with powers of 2.
As for the second one, divide the terms by the previous term and you will have your ratio.
The third requires a little mental gymnastics before the pattern will become obvious. Starting with the 2nd term, down 2, up 10, down 2, up 10, etc.
I'll let you play with the other 2.
To solve the missing numbers in these patterns, we need to identify whether they follow an arithmetic or geometric pattern and determine the common differences or common ratio. Let's go through each pattern step by step:
Pattern 1:
1, 2, __, __, __, 32
This pattern seems to be an arithmetic sequence. We can find the common difference by subtracting consecutive terms:
2 - 1 = 1
__ - 2 = 1
__ - __ = 1
So, the common difference is 1. To find the missing numbers, we can continue adding the common difference of 1 to the last known term:
2 + 1 = 3
3 + 1 = 4
4 + 1 = 5
Thus, the missing numbers in the pattern are 3, 4, 5. The common difference is 1.
Pattern 2:
800, 80, 8, 0, 8, 0.08, __, __
This pattern seems to be a combination of arithmetic and geometric sequence. Let's identify the common ratios and common differences for different parts of the pattern:
800 / 80 = 10 (ratio)
80 / 8 = 10 (ratio)
8 / 0 = undefined (division by zero defined as missing)
0 / 8 = 0 (ratio)
8 / 0.08 = 100 (ratio)
We can see that after each zero, the following number is obtained by dividing the previous number by 10 (common ratio of 10).
Using this pattern, the missing numbers can be found as follows:
0.08 / 10 = 0.008
0.008 / 10 = 0.0008
The missing numbers in the pattern are 0.008, 0.0008. The common ratio is 10.
Now, let's move to the tricky patterns:
Tricky Pattern 1:
17, 15, 25, 23, 33, 31, __, __
This pattern doesn't follow arithmetic or geometric rules but seems to alternate between two sequences. Let's analyze each sequence separately:
First sequence: 17, 25, 33...
The numbers increase by 8.
Second sequence: 15, 23, 31...
The numbers increase by 8 as well.
Therefore, the pattern alternates between two arithmetic sequences, increasing by 8. To find the missing numbers, we continue adding 8 to the last known terms:
25 + 8 = 33
33 + 8 = 41
The missing numbers in the pattern are 33 and 41.
Tricky Pattern 2:
3, 6, 7, 14, 15, 30, 31, __, __
This pattern is also a combination of arithmetic and geometric sequences. Let's separate the pattern into two parts:
First part: 3, 7, 15, 31...
The numbers seem to follow a geometric sequence, where each term is obtained by multiplying the previous term by 2, except for the first term.
Second part: 6, 14, 30...
The numbers increase by 8, similar to the tricky pattern 1.
Using these observations, we can find the missing numbers:
31 * 2 = 62
62 * 2 = 124
The missing numbers in the pattern are 62 and 124.
Tricky Pattern 3:
1, 6, 5, 10, 9, 14, 13, __, __
This pattern also alternates between two different sequences. Let's analyze it:
First sequence: 1, 5, 9, 13...
The numbers increase by 4.
Second sequence: 6, 10, 14...
The numbers increase by 4 as well.
Hence, the pattern alternates between two arithmetic sequences, increasing by 4. To find the missing numbers, we continue adding 4 to the last known terms:
13 + 4 = 17
17 + 4 = 21
The missing numbers in the pattern are 17 and 21.
By analyzing the patterns and identifying the arithmetic and geometric characteristics, we were able to find the missing numbers and the common differences or common ratio for each pattern.