Geometric Sequences Quiz

Questions and Answers

What is a distinguishing feature of a geometric sequence compared to an arithmetic sequence?

The ratio between any two consecutive terms is constant. (correct)

The sum of all terms is always greater than zero.

The first term is always the largest.

The difference between any two consecutive terms is constant.

If the first term of a geometric sequence is 3 and the common ratio is 2, what is the 4th term?

24 (correct)

How do you determine the geometric means of the sequence 2, x, 8?

By multiplying both extremes and taking the square root. (correct)

By taking the cubic root of the product.

By finding the square root of the product of the extremes.

By adding the numbers and dividing by 3.

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

<p>$S_n = a_1 \frac{1 - r^n}{1 - r}$</p> Signup and view all the answers

In a geometric sequence with a first term of 5 and a common ratio of 3, what is the 3rd term?

<p>135</p> Signup and view all the answers

Study Notes

Geometric Sequence

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 (r).
The general form of a geometric sequence can be expressed as a, ar, ar², ar³, ..., where 'a' is the first term.

Differentiating Geometric and Arithmetic Sequences

In a geometric sequence, the ratio between consecutive terms is constant (common ratio), while in an arithmetic sequence, the difference between consecutive terms is constant (common difference).
Example: In the geometric sequence 2, 6, 18, the common ratio is 3 (6/2 = 3, 18/6 = 3). In the arithmetic sequence 2, 4, 6, the common difference is 2 (4-2 = 2, 6-4 = 2).

nth Term of a Geometric Sequence

The nth term (Tn) of a geometric sequence can be calculated using the formula: Tn = a * r^(n-1), where 'a' is the first term, 'r' is the common ratio, and 'n' is the term number.

Geometric Means

Geometric means are terms that can be inserted between two known terms of a geometric sequence to maintain the ratio.
For two terms 'a' and 'b', the geometric mean (GM) can be found using the formula: GM = √(a*b).

Sum of the Terms of a Geometric Sequence

The sum of the first n terms (S_n) of a geometric sequence can be calculated using the formula:
- If r ≠ 1: S_n = a * (1 - r^n) / (1 - r)
- If r = 1: S_n = n * a, where 'n' is the number of terms.
For an infinite geometric series where |r| < 1, the sum can be calculated using: S = a / (1 - r).

Description

Test your understanding of geometric sequences with this quiz. Dive into the differences between geometric and arithmetic sequences, calculate the nth term, find geometric means, and determine the sum of terms. Perfect for students looking to reinforce their knowledge on this essential math topic.