Arithmetic and Geometric Sequences and Series Review

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 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 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 of a geometric sequence can be calculated using the formula: \( a_n = a_1 \cdot r^{(n-1)} \), 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 an arithmetic series?

Back

The sum of the first n terms of an arithmetic series can be calculated using the formula: \( S_n = \frac{n}{2} (a_1 + a_n) \) or \( S_n = \frac{n}{2} (2a_1 + (n-1)d) \).

Front

What is the formula for the sum of an infinite geometric series?

Back

The sum of an infinite geometric series can be calculated using the formula: \( S = \frac{a}{1 - r} \), where \( a \) is the first term and \( r \) is the common ratio (with \( |r| < 1 \)).

Front

Evaluate the series: 2 + 4 + 8 + ... + 64.

Back

The sum of the series is -42.

Front

Describe the sequence: -8, -6, -4, -2...

Back

This is an arithmetic sequence with a common difference of 2. The 5th term is 0, and the rule is \( a_n = 2n - 10 \).

Front

Evaluate the infinite geometric series: 1 + 1/5 + 1/25 + ...

Back

The sum of the series is \( \frac{5}{4} \).

Front

Find the 22nd term of the sequence: 5, 8, 11, ...

Back

The 22nd term is 68.