Geometric Sequence

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 common ratio in a geometric sequence?

Back

The common ratio (r) is the factor by which we multiply each term to get the next term in the sequence.

Front

If the first term a₁ of a geometric sequence is 5 and the common ratio r is 3, what is the second term a₂?

Back

a₂ = a₁ * r = 5 * 3 = 15.

Front

What does a₁ represent in a geometric sequence?

Back

a₁ represents the first term of the geometric sequence.

Front

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

Back

The nth term can be found using the formula a_n = a₁ * r^n-1.

Front

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

Back

The sum S_n of the first n terms is given by S_n = a₁ * (1 - rⁿ) / (1 - r) for r ≠ 1.

Front

If a geometric sequence has a common ratio of 2, what are the first four terms if the first term is 1?

Back

The first four terms are 1, 2, 4, 8.

Front

What does the term 'exponential growth' refer to in the context of geometric sequences?

Back

Exponential growth refers to the increase in a quantity where the rate of growth is proportional to the current value, often modeled by geometric sequences.

Front

In the sequence 3, 6, 12, 24, what is the common ratio?

Back

The common ratio is 2.

Front

If a₁ = 10 and r = -2, what is a₃?

Back

a₃ = 10 * (-2)² = 10 * 4 = 40.