Considering the jumping jacks: 50, 100, 150, 200, what is the common difference?
Now, think of this table as a set of ordered pairs. This means that the first row can be placed in an ordered pair as (1, 50). The second row can be written as (2, 100). Using this, what is the slope of the line that connects the first two points?
What is the slope of the line that connects the 3rd and 4th point?
What is the slope of the line that connects the 1st and the 4th point?
Is the common difference (aka slope aka rate of change) constant?
Why is it or is it not constant?
The common difference is 50.
The slope of the line connecting the first two points (1, 50) and (2, 100) would be (100-50)/(2-1) = 50.
The slope of the line connecting the third and fourth point (3, 150) and (4, 200) would also be 50.
The slope of the line connecting the first and fourth point (1, 50) and (4, 200) would be (200-50)/(4-1) = 50.
The common difference (slope) is constant, because the increase in the number of jumping jacks (x-value) results in a consistent increase in the number of repetitions (y-value). Each time the x-value increases by 1, the y-value increases by 50, resulting in a constant rate of change/slope.
The given series of jumping jacks is: 50, 100, 150, 200.
To find the common difference in an arithmetic sequence, we subtract consecutive terms.
Difference between 100 and 50: 100 - 50 = 50
Difference between 150 and 100: 150 - 100 = 50
Difference between 200 and 150: 200 - 150 = 50
Therefore, the common difference is 50.
Now, let's consider the series as a set of ordered pairs:
(1, 50), (2, 100), (3, 150), (4, 200).
To find the slope of a line connecting two points, we use the formula: slope = (change in y) / (change in x).
Slope between the first two points, (1, 50) and (2, 100):
change in y = 100 - 50 = 50
change in x = 2 - 1 = 1
Slope = (50) / (1) = 50.
Slope between the third and fourth points, (3, 150) and (4, 200):
change in y = 200 - 150 = 50
change in x = 4 - 3 = 1
Slope = (50) / (1) = 50.
Slope between the first and fourth points, (1, 50) and (4, 200):
change in y = 200 - 50 = 150
change in x = 4 - 1 = 3
Slope = (150) / (3) = 50.
The slopes calculated for all three scenarios are the same, which means the common difference (slope/rate of change) is constant.
The common difference is constant because each term in the sequence is obtained by continuously adding the same value, which is the common difference of 50. This constant addition results in a constant change in both the x and y values when considering the points as ordered pairs.