Arithmetic and Geometric Sequences

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.

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 explicit formula for an arithmetic sequence?

Back

The explicit formula for an arithmetic sequence is a_n = a_1 + (n-1)d, where a_1 is the first term, d is the common difference, and n is the term number.

Front

What is the explicit formula for a geometric sequence?

Back

The explicit formula for a geometric sequence is a_n = a_1 * r^(n-1), where a_1 is the first term, r is the common ratio, and n is the term number.

Front

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

Back

The common difference (d) can be found by subtracting any term from the subsequent term: d = a_(n+1) - a_n.

Front

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

Back

The common ratio (r) can be found by dividing any term by the previous term: r = a_n / a_(n-1).

Front

What is the first term of the sequence 5, 10, 15, 20...?

Back

The first term (a_1) is 5.

Front

What is the 5th term of the arithmetic sequence defined by a_n = 3 + 8(n-1)?

Back

The 5th term (a_5) is 3 + 8(5-1) = 3 + 32 = 35.

Front

What is the 4th term of the geometric sequence defined by a_n = 486(1/3)^(n-1)?

Back

The 4th term (a_4) is 486(1/3)^(4-1) = 486(1/3)^3 = 486/27 = 18.

Front

What does the term 'explicit formula' mean in the context of sequences?

Back

An explicit formula provides a direct way to calculate the nth term of a sequence without needing to know the previous terms.