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

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

Back

The common difference can be found by subtracting any term from the term that follows it. For example, in the sequence {2, 5, 8}, the common difference is 5 - 2 = 3.

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

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

Back

The common ratio can be found by dividing any term by the term that precedes it. For example, in the sequence {3, 9, 27}, the common ratio is 9 / 3 = 3.

Front

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

Back

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

Front

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

Back

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

Front

If the first term of an arithmetic sequence is 2 and the common difference is 3, what is the 5th term?

Back

The 5th term is 2 + (5-1) * 3 = 2 + 12 = 14.

Front

If the first term of a geometric sequence is 4 and the common ratio is 2, what is the 4th term?

Back

The 4th term is 4 * 2^(4-1) = 4 * 8 = 32.

Front

What is the sum of the first n terms of an arithmetic sequence?

Back

The sum 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 the sum of the first n terms of a geometric sequence?

Back

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