What is the sum of the first 200 odd numbers?
A rectangle has an area of 120 cm square we are trying to find possible perimeters. It's length and width are whole numbers. What are all the possibilities for the 2 numbers?
Give next two terms
3,4,7,11,18,29....
1,2,6,24,120....
77,49,36,18....
Odd numbers: 1 , 3 , 5...
200th od number = 399
Now:
1 + 399 = 400
3 + 397 = 400
5 + 395 = 400
....
199 + 201 = 400
100 * 400 = 40,000
A = L * W = 120
L = 120 / W
120 = 2 * 2 * 2 * 3 * 5
Ohe possibilities for the 2 numbers all numbers dividable with 2 3 and 5
W = 120 , L = 120 / 120 = 1
W = 60 , L = 2
W = 30 , L = 4
W = 15 , L = 8
W = 10 , L = 12
W = 8 , L = 15
W = 6 , L = 20
W = 4 , L = 30
W = 3 , L = 40
W = 2 , L = 60
W = 1 , L = 120
3,4,7,11,18,29....
The previous two numbers make up the next number:
3 + 4 = 7
7 + 4 = 11
11 + 7 = 18
11 + 18 = 29
Last two are:
18 + 29 = 47
and
29 + 47 = 76
3 ,4 , 7 , 11 , 18 , 29 , 47 , 76
1 , 2 , 6 , 24 , 120..
The multiplier increases by one each time:
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
120 x 6 = 720
720 x 7 = 5040
1 , 2 , 6 , 24 , 120 , 720 , 5040
77 , 49 , 36 , 18 ...
77
7 x 7 = 49
4 x 9 = 36)
3 x 6 = 18
1 x 8 = 8)
0 x 8 = 0
77 , 49 , 36 , 18 , 8 , 0
To find the sum of the first 200 odd numbers, you can use the formula for the sum of an arithmetic series.
The formula for the sum of an arithmetic series Sn = (n/2)(a + l), where Sn is the sum of the series, n is the number of terms, a is the first term, and l is the last term.
In this case, the first odd number is 1, and the last odd number is 399.
So, using the formula, we can calculate the sum:
Sn = (200/2)(1 + 399)
= 100(400)
= 40,000
Therefore, the sum of the first 200 odd numbers is 40,000.
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To find all the possible dimensions (length and width) of a rectangle with an area of 120 cm², you need to find pairs of whole numbers that multiply together to give 120.
There are several ways to find these pairs:
- You can list all the factors of 120 and look for pairs.
- You can use a factor tree to find the prime factors of 120 and then combine them to form pairs.
- You can also use trial and error by starting with 1 as the length and finding the corresponding width.
Here are all the possible pairs of whole numbers that multiply to 120:
1 x 120 = 120
2 x 60 = 120
3 x 40 = 120
4 x 30 = 120
5 x 24 = 120
6 x 20 = 120
8 x 15 = 120
10 x 12 = 120
So, the possible dimensions for the rectangle are:
- Length x Width: 1 x 120, 2 x 60, 3 x 40, 4 x 30, 5 x 24, 6 x 20, 8 x 15, 10 x 12.
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To find the next two terms in each of the given number sequences, we need to identify the pattern or rule that governs the sequence.
1. The first sequence (3, 4, 7, 11, 18, 29, ...) seems to be increasing by an increment that is constantly changing. Looking closely, we can see that the difference between each term is:
4 - 3 = 1
7 - 4 = 3
11 - 7 = 4
18 - 11 = 7
29 - 18 = 11
So, the pattern is that each term is formed by adding the previous difference to the previous term.
Therefore, the next term would be:
29 + 11 = 40
And the term after that would be:
40 + 18 = 58
So, the next two terms in the sequence are 40 and 58.
2. The second sequence (1, 2, 6, 24, 120, ...) seems to be increasing by multiplying each term by a constant factor that is constantly changing. Looking closely, we can see that each term is formed by multiplying the previous term by a number that is increasing by 1 each time:
1 x 2 = 2
2 x 3 = 6
6 x 4 = 24
24 x 5 = 120
So, the pattern is that each term is formed by multiplying the previous term by the next consecutive number.
Therefore, the next term would be:
120 x 6 = 720
And the term after that would be:
720 x 7 = 5040
So, the next two terms in the sequence are 720 and 5040.
3. The third sequence (77, 49, 36, 18, ...) seems to be decreasing, but the pattern is not immediately obvious. By analyzing the differences between each term, we can see that they are decreasing in a particular way:
77 - 49 = 28
49 - 36 = 13
36 - 18 = 18
So, the differences are not consistent. However, if we look at the differences of the differences, we can see that they are constant:
28 - 13 = 15
Therefore, we can assume that the pattern is a quadratic sequence.
Using this pattern, the next difference would be:
18 - 15 = 3
And the term after that would be:
18 - 3 = 15
So, the next two terms in the sequence are 18 and 15.