AP Precalculus Unit 2A: 2.1 (Change in Arithmetic and Geometric Sequences) and 2.2 (Change in Linear and Exponential Functions)

These flashcards cover key concepts of exponential and logarithmic functions, arithmetic and geometric sequences as discussed in the lecture notes.

sequence

An ordred list of numbers that could be finite or infinite. Each listed number is a term. A graph of a sequence contains discrete points, not a connected line or curve.

arithmetic sequence

A sequence in which each successive term has a common difference (or a constant rate of change).

equation for the nth term of an arithmetic sequence

a_n = a₀ + d_n, where a₀ is the first term and d is the common difference.

equation for an arithmetic sequence using ANY term (kth term)

a_n = a_k + d(n-k), where a_k is the kth term of the sequence.

geometric sequence

A sequence in which each successive term has a common ratio (or constant proportional change).

equation for the nth term of a geometric sequence

g_n = g₀ r^n, where g₀ is the initial value (zero term) and r is the common ratio

equation for an geometric sequence using ANY term (kth term)

g_n = g_kr^(n-k), where g_k is the kth term of the sequence.

Difference in growth between arithmetic and geometric sequences

Increasing arithmetic sequences increase equally with each step, while increasing geometric sequences increase by a larger amount with each successive step.

Similarities of linear functions f(x) = b+mx and arithmetic sequences a_n = a₀ + d_n

Both can be expressed as an initial value (b or a₀) plus repeated addition of a constant rate of change, the slope (m or d).

Similarities of linear functions and arithmetic sequences when expressed using a kth term / point? 

1. Arithmetic sequences for the kth term use a_n = a_k + d(n-k), which are based on a known difference (d), and a kth term.

2. Linear functions can be expressed in the form f (x) = y₁ + m(x-x_1), based on a known slope (m), and a point, (x₁, y₁).  Equation of a line = y-y₁ = m (x-x₁)

3. The point (x₁, y₁) for a linear function is similar to the term (k, a_k) of an arithmetic sequence.

Similarities of exponential functions f(x) = ab^x and geometric sequences g_n = g₀rⁿ

Both can be expressed as an initial value (a or g₀) times repeated multiplication by a constant proportion (b or r).

Similarities of exponential functions and geometric sequences when expressed using a kth term / point

1. Geometric sequences use the equation: g_n = g_kr^(n-k), which are based on a known ratio ( r ) and a kth term.

2. The shifted exponential function can be expressed using the equation: f(x) = y₁r^x-x₁ based on a known ratio( r ) and a point (x₁ y₁).

3. The point (x₁, y₁) for the exponential funtion is similar to the term (k, g_k)

True or False? The domain of a sequence is always the same as the domain of its corresponding function.

False; Sequences and their corresponding functions may have different domains.

Linear functions and exponential functions can be expressed analytically in terms of an initial value and a constant involved in change. There is a difference between the two though, what is that difference?

Linear functions are based on addition and exponential functions are based on multiplication.

Over equal-length input-value intervals, if the output values of a function change at constant rate, the function is _______.

Linear (Adding the slope)

Over equal-length input-value intervals, if the output values of a function change proportionally, the function is _______.

Exponential (Multiplying the ratio)

If you know a function is linear or exponential (for sequences that would be arithmetic or geometric), what do you need to come up with an equation (rule) for the function or sequence?

Two distinct sequence or function values