HW#30 Explicit and Recursive Formulas (honors)

Front

What is an arithmetic sequence?

Back

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (d).

Front

How do you find the common difference in an arithmetic sequence?

Back

To find the common difference (d), subtract any term from the subsequent term: d = a_{n+1} - a_n.

Front

What is a geometric sequence?

Back

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio (r).

Front

How do you find the common ratio in a geometric sequence?

Back

To find the common ratio (r), divide any term by the previous term: r = a_{n+1} / a_n.

Front

What is an explicit formula for a sequence?

Back

An explicit formula defines the nth term of a sequence directly in terms of n. For example, for an arithmetic sequence: a_n = a_1 + (n-1)d.

Front

What is a recursive formula for a sequence?

Back

A recursive formula defines each term of a sequence based on the previous term(s). For example, for an arithmetic sequence: a_n = a_{n-1} + d.

Front

Given the explicit formula a_n = 5(2)^{n-1}, what is a_5?

Back

a_5 = 5(2)^{5-1} = 5(16) = 80.

Front

What are the first four terms of the sequence defined by a_n = 4(5)^{n-1}?

Back

The first four terms are: a_1 = 4, a_2 = 20, a_3 = 100, a_4 = 500.

Front

Identify the type of sequence: 3, 6, 9, 12...

Back

This sequence is an arithmetic sequence because the common difference is 3.

Front

What is the next term in the sequence: 2, -4, 8, -16?

Back

The next two terms are 32 and -64, following the pattern of multiplying by -2.