Definition of Geometric Sequences
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 ratio. Simply, each subsequent term is a product of the previous term and a consistent value. For a comparison, see the basics of arithmetic sequences.
Understanding the Definition
- Geometric Sequence: A sequence like 3, 6, 12, 24, ... where each term after the first is obtained by multiplying the previous one by a fixed, non-zero number.
- Common Ratio: The fixed number we multiply by, often denoted as "r".
Example 1: Identifying Geometric Sequences
Consider the sequence: 5, 10, 20, 40, ...
- The ratio between consecutive terms (10/5, 20/10, 40/20, ...) is always 2.
- Therefore, this is a geometric sequence with a common ratio of 2.
Example 2: Non-Example
Consider the sequence: 2, 4, 7, 11, ...
- The ratio between consecutive terms is not constant.
- Thus, this is not a geometric sequence.
Common Ratio
The common ratio, denoted as "r", is the factor by which we multiply one term to get the next in a geometric sequence. It is found by dividing any term (an) by its preceding term (a(n-1)).
r = an / a(n-1)
For more on related topics, you might explore exponential equations.
Example: Finding the Common Ratio
Consider the sequence: 3, 6, 12, 24, ...
- To find the common ratio, divide the second term by the first, or the third by the second, etc.
- r = 6 / 3 = 2
Nth Term Formula
The nth term of a geometric sequence can be found using the formula:
an = a1 * r(n-1)
where:
- an is the nth term,
- a1 is the first term,
- r is the common ratio,
- n is the term number.
Understanding sequences can also help with topics like the introduction to sigma notation.
Example: Finding the Nth Term
Consider the sequence: 4, 8, 16, 32, ...
- Here, a1 = 4 and r = 2.
- To find the 10th term (a10): a10 = a1 * r(10-1) a10 = 4 * 29 a10 = 4 * 512 a10 = 2048
Infinite and Partial Sum Formulas
Geometric sequences can also be summed, and there are formulas to find both the infinite sum and the sum of the first n terms (partial sum).
Infinite Sum Formula
If |r| < 1, the infinite sum is given by:
S = 1 / (1 - r)
Partial Sum Formula
The sum of the first n terms, Sn, is given by:
Sn = (a1 * (1 - rn)) / (1 - r)
For more on proofs, you can look into direct and indirect proofs.
Example: Using Sum Formulas
Consider the sequence: 1, 1/2, 1/4, 1/8, ...
- Here, a1 = 1 and r = 1/2.
- The infinite sum S is: S = 1 / (1 - (1/2)) S = 2
- The sum of the first 4 terms S4 is: S4 = (1 * (1 - (1/2)4)) / (1 - 1/2) S4 = (1 * (1 - 1/16)) / (1/2) S4 = (15/16) / (1/2) S4 = 30/16
Applications in Exam Questions
Question 1: Finding a Specific Term
Consider the geometric sequence: 5, 10, 20, 40, ...
Find the 7th term of the sequence.
Solution:
- First term, a1 = 5
- Common ratio, r = 2
- Using the nth term formula: a7 = a1 * r(7-1) a7 = 5 * 26 a7 = 5 * 64 a7 = 320
Question 2: Identifying the Common Ratio
The 3rd term in a geometric sequence is 27, and the 6th term is 243. Find the common ratio.
Solution:
- Using the nth term formula, we have two equations: a3 = a1 * r(3-1) = 27 a6 = a1 * r(6-1) = 243
- Dividing the second equation by the first to find r: r3 = 243 / 27 r3 = 9 r = 3
For additional practice with sequences, see the basics of probability.
Question 3: Forming a Geometric Sequence
Create a geometric sequence with the first term of 3 and a common ratio of 4.
Solution:
- Using the formula, we can find the next few terms: a2 = a1 * r = 3 * 4 = 12 a3 = a2 * r = 12 * 4 = 48 a4 = a3 * r = 48 * 4 = 192
- Thus, the sequence is: 3, 12, 48, 192, …
FAQ
Absolutely, geometric sequences are widely used to model various real-world phenomena, particularly those that exhibit exponential growth or decay. For instance, in finance, geometric sequences can model compound interest, where the amount of money grows exponentially over time. In biology, they can represent population growth under ideal conditions. The common ratio then becomes a crucial factor, representing the rate of growth or decay, and understanding how to manipulate and interpret geometric sequences becomes vital in making accurate predictions and informed decisions in various fields.
Yes, a geometric sequence can have a common ratio of 1 or 0, but these cases are somewhat trivial. If the common ratio, r, is 1, every term in the sequence will be equal to the first term, creating a constant sequence. If r is 0, every term after the first will be 0, regardless of the value of the first term. These scenarios might not be very interesting in the context of exploring geometric growth or decay but are technically valid geometric sequences and might have applications in certain mathematical problems or models.
Geometric sequences and exponential functions are intrinsically linked. A geometric sequence is essentially a discrete version of an exponential function. The nth term formula for a geometric sequence, an = a1 * r(n-1), resembles the formula for exponential growth, y = a * bx, where a is the initial amount, b is the growth factor, and x is the exponent. In the context of the geometric sequence, n-1 acts as the exponent, making the sequences a stepwise model of exponential growth or decay. This relationship allows concepts and models to be transferred between continuous and discrete contexts, providing a versatile analytical framework.
The geometric mean is closely related to geometric sequences as it represents the central tendency of a set of numbers by using the product of their values. Specifically, the geometric mean of n numbers is equal to the nth root of the product of the numbers. In the context of a geometric sequence, the geometric mean of any two consecutive terms is equal to the common ratio, r. This is because when you multiply two consecutive terms and take the square root (since n=2), you are essentially finding the common ratio. This property is unique to geometric sequences and is utilised in various mathematical and statistical applications.
Changing the common ratio in a geometric sequence significantly impacts the progression of the terms. If the common ratio is greater than 1, the terms will increase exponentially. If it is between 0 and 1, the terms will decrease and approach zero. A negative common ratio will cause the terms to alternate in sign. Understanding the impact of the common ratio is crucial as it dictates the behaviour of the sequence, influencing whether it diverges or converges, and thus affecting the sum of its terms and its applications in various mathematical contexts.
Practice Questions
The 8th term of a geometric sequence can be found using the formula for the nth term: an = a1 * r(n-1), where a1 is the first term, r is the common ratio, and n is the term number. For this sequence, a1 = 3 and r = 2 (since 6/3 = 2). Substituting these values and n = 8 into the formula, we get: a8 = 3 * 2(8-1) = 3 * 27 = 3 * 128 = 384. Therefore, the 8th term of the sequence is 384.
To find the sum Sn of the first n terms of a geometric sequence, we can use the formula: Sn = a1 * (1 - rn) / (1 - r), where a1 is the first term, r is the common ratio, and n is the number of terms. For this sequence, a1 = 5, r = 3, and n = 6. Plugging these values into the formula, we get: S6 = 5 * (1 - 36) / (1 - 3) = 5 * (1 - 729) / (-2) = 5 * 365 = 1825. Thus, the sum of the first 6 terms of the sequence is 1825.
