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.

Tags

CCSS.HSF.BF.A.2

Front

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

Back

The common ratio (r) can be found by dividing any term in the sequence by the previous term, i.e., r = a_n / a_(n-1).

Tags

CCSS.HSF.BF.A.2

Front

What is an arithmetic sequence?

Back

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

Tags

CCSS.HSF.BF.A.2

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, i.e., d = a_n - a_(n-1).

Tags

CCSS.HSF.BF.A.2

Front

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

Back

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

Tags

CCSS.HSA.SSE.B.4

Front

What is the formula for the sum of the first n terms of an arithmetic series?

Back

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

Front

What is the significance of the first term in a geometric series?

Back

The first term (a_1) is crucial as it serves as the starting point for the series and is multiplied by the common ratio to generate subsequent terms.

Tags

CCSS.HSA.SSE.B.4

Front

What happens to the sum of a geometric series if the common ratio is greater than 1?

Back

If the common ratio is greater than 1, the sum of the series diverges to infinity as more terms are added.

Tags

CCSS.HSA.SSE.B.4

Front

What is the behavior of the sum of a geometric series if the common ratio is between 0 and 1?

Back

If the common ratio is between 0 and 1, the sum converges to a finite value as more terms are added.

Front

How do you determine if a series converges or diverges?

Back

A series converges if the sum approaches a finite limit as the number of terms increases; it diverges if the sum increases without bound.