Explore the relationship between linear functions and arithmetic sequences to find which linear function represents the arithmetic sequence 60, 50, 40…..
An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. In other words, the difference between any two consecutive terms is the same. The formula to find the nth term of an arithmetic sequence is given by:
a_n = a_1 + (n-1)d
where:
a_n is the nth term,
a_1 is the first term,
d is the common difference,
and n is the term number.
In this problem, the arithmetic sequence is 60, 50, 40,... We can see that the common difference between each consecutive term is -10 (60 - 50 = -10, 50 - 40 = -10).
To find the linear function that represents this arithmetic sequence, we need to find the slope (m) and y-intercept (b) of the corresponding linear function in the form y = mx + b.
The slope of a linear function can be calculated using the formula:
m = (y2 - y1) / (x2 - x1)
Let's take the first two terms (60, 50) to find the slope:
m = (50 - 60) / (2 - 1) = -10
Now that we have the slope, we can write the linear function in slope-intercept form:
y = mx + b
y = -10x + b
To find the y-intercept, we can substitute the coordinates of one point on the line (e.g., (1, 60)):
60 = -10(1) + b
60 = -10 + b
b = 70
Therefore, the linear function that represents the arithmetic sequence 60, 50, 40, ... is:
y = -10x + 70