Flashcard Review for series and sequence

Front

What is a sequence in mathematics?

Back

A sequence is an ordered list of numbers that follow a specific pattern or rule.

Front

Define an arithmetic sequence.

Back

An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant.

Front

What is the formula for the nth term of an arithmetic sequence?

Back

The nth term (a_n) can be calculated using the formula: a_n = a_1 + (n-1)d, where a_1 is the first term and d is the common difference.

Front

How do you find the sum of the first n terms of an arithmetic series?

Back

The sum (S_n) of the first n terms can be calculated using the formula: S_n = n/2 * (a_1 + a_n), where a_n is the nth term.

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.

Front

What is the formula for the nth term of a geometric sequence?

Back

The nth term (a_n) can be calculated using the formula: a_n = a_1 * r^(n-1), where a_1 is the first term and r is the common ratio.

Front

How do you find the sum of the first n terms of a geometric series?

Back

The sum (S_n) of the first n terms can be calculated using the formula: S_n = a_1 * (1 - r^n) / (1 - r), for r ≠ 1.

Front

What is a recursive sequence?

Back

A recursive sequence is a sequence in which each term is defined as a function of one or more of the preceding terms.

Front

Provide an example of a recursive formula for an arithmetic sequence.

Back

An example is: a_n = a_(n-1) + d, where d is the common difference.

Front

What is the first term of the sequence defined by a_n = 2(5)^n?

Back

The first term (when n=1) is a_1 = 2(5)^1 = 10.