Algebra 2 Midterm

A recursive formula defines each term of a sequence using the previous term(s). For example, in the sequence 81, -27, 9, -3, 1, the recursive formula is $$a_1=81; a_n=a_{n-1} \cdot (-\frac{1}{3})$$ .

To find the equation of a line given two points (x1, y1) and (x2, y2), first calculate the slope (m) using $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ . Then use the point-slope form: $$y - y_1 = m(x - x_1)$$ .

An arithmetic sequence has a constant difference between consecutive terms (common difference, d), while a geometric sequence has a constant ratio between consecutive terms (common ratio, r).

To determine if a sequence is arithmetic, check if the difference between consecutive terms is constant. For geometric sequences, check if the ratio of consecutive terms is constant.

The common difference (d) in an arithmetic sequence is the fixed amount added to each term to get the next term. For example, in the sequence 3, 6, 9, 12, d = 3.

The common ratio (r) in a geometric sequence is the fixed factor multiplied to each term to get the next term. For example, in the sequence 3, 6, 12, 24, r = 2.

The point-slope form of a linear equation is given by $$y - y_1 = m(x - x_1)$$ , where m is the slope and (x1, y1) is a point on the line.

To solve a system of equations, you can use substitution, elimination, or graphing methods to find the values of the variables that satisfy all equations simultaneously.

The formula for the slope (m) of a line passing through two points (x1, y1) and (x2, y2) is $$m = \frac{y_2 - y_1}{x_2 - x_1}$$ .

A sequence is neither arithmetic nor geometric if it does not have a constant difference or a constant ratio between consecutive terms.