Geometric Progression Basics

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What is a geometric progression?

A geometric progression is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio.

How is the common ratio (r) in a geometric progression calculated?

The common ratio is calculated by dividing any term in the sequence by the previous term.

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

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

How does a geometric sequence differ from an arithmetic sequence?

In a geometric sequence, each term is found by multiplying the previous one by a fixed number, while in an arithmetic sequence, a fixed number is added to the previous term.

What are finite and infinite geometric sequences?

A finite geometric sequence has a limited number of terms, while an infinite geometric sequence continues indefinitely.

Can a geometric sequence have negative terms?

Yes, a geometric sequence can have negative terms if the common ratio is negative.

What happens if the common ratio in a geometric progression is equal to 1?

If the common ratio is 1, all terms in the sequence will be the same as the first term.

Give an example of a geometric progression with a common ratio of 2.

An example is the sequence 3, 6, 12, 24, where the first term is 3 and the common ratio is 2.

What is a geometric series?

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

How does the sum of an infinite geometric series differ from a finite one?

An infinite geometric series can converge to a finite sum if the common ratio is between -1 and 1, whereas a finite series always has a definite sum.

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

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

What condition must a geometric series meet to converge?

For an infinite geometric series to converge, the absolute value of the common ratio must be less than 1 (|r| < 1).

How can geometric progressions be applied in finance?

Geometric progressions are used to model compound interest and exponential growth in financial calculations.

What is the significance of the zero term in a geometric sequence?

In a geometric sequence, having a zero term means that subsequent terms will all be zero if the common ratio is non-zero.

In a geometric sequence with a common ratio of 1/2 and a first term of 8, what is the 4th term?

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




Test Your Knowledge

Select the correct option


1. What happens if the common ratio in a geometric progression is equal to 1?

All terms in the sequence will be the same as the first term.

The sequence becomes an arithmetic sequence.

The sequence will diverge.

The sequence will become finite.

2. What is the formula for the nth term of a geometric progression?

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

a_n = a * r^n, where a is the first term and r is the common ratio.

a_n = a + rn, where a is the first term and r is the common difference.

a_n = n * a * r, where a is the first term and r is the common ratio.

3. How does the sum of an infinite geometric series differ from a finite one?

An infinite geometric series can converge to a finite sum if the common ratio is between -1 and 1.

An infinite geometric series will always diverge.

An infinite geometric series is always less than any finite series.

An infinite series converges only if the common ratio is 0.

4. What condition must a geometric series meet to converge?

The absolute value of the common ratio must be less than 1 (|r| < 1).

The common ratio must be greater than 1.

The common ratio must be between 1 and 2.

The absolute value of the common ratio must be greater than 1.

5. How can geometric progressions be applied in finance?

They are used to model compound interest and exponential growth.

They are used to calculate the average rate of return.

They model linear depreciation of assets.

They determine fixed savings plans.

6. What is a geometric series?

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

A geometric series is any series that involves cubic powers.

A geometric series is a sequence where each term is incremented by a fixed amount.

A geometric series is a sequence where every fifth term is multiplied by a constant.

7. In a geometric sequence with a common ratio of 1/2 and a first term of 8, what is the 4th term?

1

2

4

0.5

8. What is the significance of the zero term in a geometric sequence?

Subsequent terms will all be zero if the common ratio is non-zero.

It marks the end of the sequence.

It causes the sequence sum to be infinite.

It doubles the value of the following terms.