Question

Relate the defining characteristics of an arithmetic sequence (common difference and first term) with those of a linear equation (slope and y-intercept). Consider how a point slope form equation and the explicit form of an arithmetic sequence can use the same inputs to be created. Provide an example to support your wriiting.

Answer 1

The common difference in an arithmetic sequence is analogous to the slope of a linear equation, and the first term of an arithmetic sequence is like the y-intercept of a linear equation. For example, an arithmetic sequence with a first term of 2 and a common difference of 3 can be represented by the linear equation y = 3x + 2, where 3 is the slope and 2 is the y-intercept. Both concepts depict a constant rate of growth.

Step-by-step explanation:

The defining characteristics of an arithmetic sequence and those of a linear equation are closely related. An arithmetic sequence is a sequence of numbers with a constant difference between each consecutive term. This constant difference is analogous to the slope of a linear equation in the form y = mx + b, where 'm' is the slope, and 'b' is the y-intercept. The first term of an arithmetic sequence is like the y-intercept of a linear equation because it represents the starting point of the sequence or line at x=0.

To illustrate this with an example, consider an arithmetic sequence that starts with a first term of 2 and has a common difference of 3. The nth term of this sequence, denoted a_n, can be found using the formula a_n = a_1 + (n - 1)d, where d is the common difference and a_1 is the first term. For our sequence, the explicit form is a_n = 2 + (n - 1)×3.

In a linear context, this corresponds to a linear equation y = 3x + 2, where the slope is 3 (rise of 3 over a run of 1) and the y-intercept is 2. In both the arithmetic sequence and the linear equation, the rate of growth is constant, which is why the graph of an arithmetic sequence, plotted with the sequence term number along the x-axis and sequence value along the y-axis, will form a straight line indicative of linear behaviour.

Answer 2

Arithmetic sequences and linear equations are structurally similar: an arithmetic sequence's common difference directly corresponds to the slope of a linear equation and its first term to the linear equation's y-intercept. An example comparing these concepts is the arithmetic sequence 3, 6, 9, 12, ... and the linear equation y = 3x.

Step-by-step explanation:

Arithmetic sequences and linear equations share a fundamental similarity in their structures and characteristics. Specifically, the common difference in an arithmetic sequence is analogous to the slope of a linear equation, while the first term of an arithmetic sequence corresponds to the y-intercept of a linear equation. In an arithmetic sequence, the nth term is calculated using the formula an = a1 + (n-1)d, where a1 is the first term and d is the common difference. When we compare this to a linear equation in slope-intercept form y = mx + b, we notice the parallel with m representing the slope and b representing the y-intercept. Therefore, an arithmetic sequence can be thought of as the discrete version of a linear equation.

For example, consider the arithmetic sequence 3, 6, 9, 12, ... which has a common difference of 3 and a first term of 3. This sequence can be represented by the explicit formula an = 3 + (n-1)(3) = 3n. Similarly, a linear equation with a slope of 3 and a y-intercept of 0 is y = 3x. Both represent the same rate of change, with the arithmetic sequence describing this change in terms of positions in the sequence and the linear equation showing it graphically as a straight line on a Cartesian plane.