Sequences and Series

Front

What is a sequence?

Back

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

Front

What is a series?

Back

A series is the sum of the terms of a sequence.

Front

Define an explicit rule for a sequence.

Back

An explicit rule is a formula that allows you to find the nth term of a sequence directly.

Front

What is the formula for the sum of the first n terms of a geometric series?

Back

S_n = a(1 - r^n) / (1 - r), where a is the first term and r is the common ratio.

Front

What does it mean for a sequence to converge?

Back

A sequence converges if its terms approach a specific value as n increases.

Front

What does it mean for a sequence to diverge?

Back

A sequence diverges if its terms do not approach any specific value as n increases.

Front

What is the common ratio in a geometric sequence?

Back

The common ratio is the factor by which each term of the sequence is multiplied to get the next term.

Front

How do you find the nth term of a geometric sequence?

Back

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

Front

What is the first term of the sequence defined by a_n = 7(-1)^(n-1)?

Back

The first term (n=1) is a_1 = 7(-1)^(1-1) = 7.

Front

Calculate the sum of the first five terms of the geometric sequence with a = 2 and r = 3.

Back

S_5 = 2(1 - 3^5) / (1 - 3) = 2(1 - 243) / (-2) = 242.