6 (a) Here Are The First Five Terms Of An Arithmetic Sequence: ${ 4, 9, 14, 19, 24 }$Find, In Terms Of $n$, An Expression For The $ N N N T H Th T H Term Of This Sequence.(b) An Expression For The $n T H Th T H Term Of
An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. In this problem, we are given the first five terms of an arithmetic sequence: 4, 9, 14, 19, 24. We need to find an expression for the nth term of this sequence in terms of n.
Understanding the Pattern
To find the nth term of the sequence, we need to understand the pattern of the sequence. Let's examine the differences between consecutive terms:
- 9 - 4 = 5
- 14 - 9 = 5
- 19 - 14 = 5
- 24 - 19 = 5
As we can see, the difference between consecutive terms is constant, which is 5. This means that the sequence is an arithmetic sequence with a common difference of 5.
Finding the nth Term
Now that we have identified the common difference, we can use the formula for the nth term of an arithmetic sequence:
an = a1 + (n - 1)d
where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
In this case, a1 = 4 (the first term), d = 5 (the common difference), and n is the term number. Plugging these values into the formula, we get:
an = 4 + (n - 1)5
Simplifying the expression, we get:
an = 4 + 5n - 5
an = 5n - 1
Therefore, the expression for the nth term of this sequence is 5n - 1.
Example
Let's use the expression we found to find the 10th term of the sequence.
a10 = 5(10) - 1 = 50 - 1 = 49
So, the 10th term of the sequence is 49.
6 (b) Finding the nth Term of a Geometric Sequence
A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. In this problem, we are given the first five terms of a geometric sequence: 2, 6, 18, 54, 162. We need to find an expression for the nth term of this sequence in terms of n.
Understanding the Pattern
To find the nth term of the sequence, we need to understand the pattern of the sequence. Let's examine the ratios between consecutive terms:
- 6 / 2 = 3
- 18 / 6 = 3
- 54 / 18 = 3
- 162 / 54 = 3
As we can see, the ratio between consecutive terms is constant, which is 3. This means that the sequence is a geometric sequence with a common ratio of 3.
Finding the nth Term
Now that we have identified the common ratio, we can use the formula for the nth term of a geometric sequence:
an = a1 * r^(n - 1)
where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.
In this case, a1 = 2 (the first term), r = 3 (the common ratio), and n is the term number. Plugging these values into the formula, we get:
an = 2 * 3^(n - 1)
Simplifying the expression, we get:
an = 2 * 3^n / 3
an = 2 * 3^n / 3
an = 2 * 3^(n - 1)
Therefore, the expression for the nth term of this sequence is 2 * 3^(n - 1).
Example
Let's use the expression we found to find the 10th term of the sequence.
a10 = 2 * 3^(10 - 1) = 2 * 3^9 = 2 * 19683 = 39366
So, the 10th term of the sequence is 39366.
Conclusion
Q: What is an arithmetic sequence?
A: An arithmetic sequence is a sequence of numbers in which the difference between any two consecutive terms is constant. For example, the sequence 2, 5, 8, 11, 14 is an arithmetic sequence because the difference between consecutive terms is 3.
Q: What is a geometric sequence?
A: A geometric sequence is a sequence of numbers in which each term after the first is found by multiplying the previous term by a fixed, non-zero number called the common ratio. For example, the sequence 2, 6, 18, 54, 162 is a geometric sequence because each term is obtained by multiplying the previous term by 3.
Q: How do I find the nth term of an arithmetic sequence?
A: To find the nth term of an arithmetic sequence, you can use the formula:
an = a1 + (n - 1)d
where an is the nth term, a1 is the first term, n is the term number, and d is the common difference.
Q: How do I find the nth term of a geometric sequence?
A: To find the nth term of a geometric sequence, you can use the formula:
an = a1 * r^(n - 1)
where an is the nth term, a1 is the first term, r is the common ratio, and n is the term number.
Q: What is the difference between an arithmetic sequence and a geometric sequence?
A: The main difference between an arithmetic sequence and a geometric sequence is the way the terms are related. In an arithmetic sequence, the difference between consecutive terms is constant, while in a geometric sequence, each term is obtained by multiplying the previous term by a fixed number.
Q: Can I have a sequence that is both arithmetic and geometric?
A: Yes, it is possible to have a sequence that is both arithmetic and geometric. For example, the sequence 2, 6, 12, 20, 30 is both an arithmetic sequence (with a common difference of 4) and a geometric sequence (with a common ratio of 3).
Q: How do I determine if a sequence is arithmetic or geometric?
A: To determine if a sequence is arithmetic or geometric, you can look at the differences or ratios between consecutive terms. If the differences are constant, the sequence is arithmetic. If the ratios are constant, the sequence is geometric.
Q: Can I use the formulas for arithmetic and geometric sequences to find the nth term of any sequence?
A: Yes, you can use the formulas for arithmetic and geometric sequences to find the nth term of any sequence, as long as the sequence is either arithmetic or geometric.
Q: What are some real-world applications of arithmetic and geometric sequences?
A: Arithmetic and geometric sequences have many real-world applications, including:
- Finance: Compound interest and annuities
- Music: Frequency and pitch
- Science: Population growth and decay
- Engineering: Signal processing and filtering
Q: Can I use arithmetic and geometric sequences to model real-world phenomena?
A: Yes, you can use arithmetic and geometric sequences to model real-world phenomena, such as population growth, compound interest, and signal processing.
Q: Are there any other types of sequences besides arithmetic and geometric sequences?
A: Yes, there are other types of sequences, including:
- Harmonic sequences
- Fibonacci sequences
- Recursive sequences
Each of these types of sequences has its own unique properties and applications.