Explicit Formula
Introduction
Explicit formulas provide a convenient way to express the terms of a sequence using a single formula. For instance, in an arithmetic sequence, each term can be calculated using the formula `a_n = a_1 + (n - 1)d`, where `a_n` represents the `n`th term, `a_1` represents the first term, `d` is the common difference between consecutive terms, and `n` indicates the term's position in the sequence. This explicit equation allows us to find any term in the sequence without needing to know all the other terms. In general, explicit formulas can be used for arithmetic, geometric, or harmonic sequences, allowing us to calculate the `n`th term directly by plugging in the value of `n`.
Understanding Explicit Formulas
In math, explicit formulas are handy tools for expressing the terms of a sequence without needing to list them all out. Essentially, these formulas give you a direct way to find any term in a sequence without having to know the previous ones. Each sequence can have its own unique formula that represents all its terms.
For instance, in an arithmetic sequence, where each term increases by a fixed amount, you can use the formula ` a_n = a_1 + (n - 1) \cdot d `, where `a_n` is the `n`th term, ` a_1` is the first term and ` d ` is the common difference between terms.
Similarly, in a geometric sequence, where each term is found by multiplying the previous one by a fixed ratio, the formula is ` a_n = a_1 \cdot r^{(n - 1)} `, with `a_n` being the `n`th term, ` a_1 ` being the first term and ` r ` the common ratio.
Then, there's the harmonic sequence, where each term is the reciprocal of an arithmetic sequence. Its formula is ` a_n = \frac{1}{a_1 + (n - 1) \cdot d} `, with `a_n` being the `n`th term, ` a_1 ` being the first term and ` d ` the common difference of the corresponding arithmetic sequence.
These explicit formulas are helpful because they let you find any term of these sequences by just plugging in the value of ` n `.
Steps to Find the Explicit Formulas
Lets see how to find explicit formula of an arithmetic sequence by following these three simple steps.
Step `1`: Determine the first term `(a_1)` and the common difference `(d)` of the sequence.
Step `2`: Substitute these values into the formula for the `n`th term:
\(a_n = a_1 + (n - 1)d \)
Step `3`: Simplify the expression to get the explicit formula.
Similar steps can be followed to find explicit formulas for geometric and harmonic progressions.
Explicit Formula for Arithmetic Sequence
The explicit formula or explicit rule for an arithmetic sequence helps find any term without needing to calculate all the previous ones. The formula goes like this: `a_n = a_1 + (n - 1)d`.
Example: Write the explicit formula for the sequence `3, 6, 9, 12, 15, …`
Solution:
For the sequence `3, 6, 9, 12, 15, ...` each time we add `3` to get the next term.
Here, `a_1 = 3` (that's our first term), and the common difference (denoted as `d`) is also `3`.
To find the `n`th term, we plug in the values: `a_n = 3 + (n - 1)3`.
This simplifies to `a_n = 3 + 3n-3`, or more neatly as `a_n = 3n`.
Now we can find any term in the sequence without doing all the math from the beginning.
Explicit Formula for Geometric Sequence
The explicit formula for a geometric sequence helps find any term in the sequence. The formula looks like this: `a_n = a_1 \cdot r^{n-1}`, where `a_1` is the first term and `r` is the common ratio. This common ratio is what you multiply each term by to get the next one. So, if you know `a_1` and `r`, you can find any term you want.
Example: Write the explicit formula for the sequence `4,12,36,108, ...`
Solution:
For the sequence `4, 12, 36, 108, ...` each time we multiply `3` to get the next term.
Here, `a_1 = 4` (that's our first term), and the common ratio (denoted as `r`) is `3` because `r = \frac{9}{3} = 3`. Now, plug these into our formula: `a_n = 4 \cdot 3^{n-1}`. This gives us the nth term of the sequence.
Now you can find any term in the sequence without doing all the math from the beginning.
Explicit Formula for Harmonic Sequence
A harmonic sequence is a sequence of numbers where each term is the reciprocal of an arithmetic progression of integers. In simpler terms, it's a sequence where each term is the reciprocal of a series of evenly spaced numbers.
The explicit formula for the harmonic sequence is a very helpful to quickly find any term in the sequence without having to calculate all the preceding terms. Each term in the harmonic sequence follows a pattern: `1/a_1`, `1/(a_1 + d)`, `1/(a_1 + 2d)`, and so on, up to `1/(a_1 + (n - 1)d)`, where `a_1` is the first term and `d` is the common difference between terms. This formula, `a_n = 1/(a_1 + (n - 1)d)`, allows you to directly determine the `n`-th term in the sequence without needing to know the terms that come before it.
Example: Write the explicit formula for the sequence `1/3, 1/7, 1/11, 1/15, ...`
Solution:
For the sequence `1/3, 1/7, 1/11, 1/15,...` the denominators are in an arithmetic sequence.
Here, the first term `a_1` is 3, and the common difference `d` (between the denominators) is `7 - 3 = 4`.
Using the explicit formula, we can find the `n`-th term as `a_n = 1/(3 + (n - 1)4)`.
Key Points Regarding Explicit Formulas
- It's not always possible to describe a set of terms using a straightforward explicit formula.
- If there's a clear pattern that each term follows in a sequence then you can come up with what's called an explicit formula.
- Apart from the explicit formulas of arithmetic, geometric, and harmonic sequences, there can be any other formulas. For instance, take the sequence `2, 6, 12, 20, 30, ...` Each term is the result of multiplying its position by one more than the position, so the explicit formula for this sequence is ` a_n = n(n + 1) `. `1, 4, 9, 16, 25, ....`. Each term in this sequence is the square of its position in the sequence, which can be represented by the formula `a_n = n^2`.
- In addition to explicit formula for any sequence, the `n`-th term of a sequence can also be found using recursive formula. The recursive formula of a sequence uses the value of its `n-1`the term to find the value of its `n`th term. Explicit and recursive formula are used intermittently depending on the known values of a sequence.
Solved Examples
Example `1`: Find the `10`th term of the arithmetic sequence where the first term `a_1 = 3` and the common difference `d = 2`.
Solution:
`a_1 = 3`
`d = 2`
Explicit equation formula for an arithmetic sequence is:
`a_n = a_1 + (n - 1)d`
`a_{10} = 3 + (10 - 1) \times 2`
`= 3 + 9 \times 2`
`= 3 + 18`
`= 21`
Therefore, the `10`th term of the sequence is `a_{10} = 21`.
Example `2`: Determine the `7`th term of the geometric sequence where the first term `a_1 = 2` and the common ratio `r = 3`.
Solution:
`a_1 = 2`
`r = 3`
Explicit formula for a geometric sequence is:
`a_n = a_1 \times r^{(n-1)}`
`a_7 = 2 \times 3^{(7-1)}`
`= 2 \times 3^6`
`= 2 \times 729`
`= 1458`
Therefore, the `7`th term of the sequence is `a_7 = 1458`.
Example `3`: Find the `5`th term of the sequence `1, 4, 9, 16, ...`.
Solution:
The given sequence is `1, 4, 9, 16, ...`.
The given sequence can be written as: `1^2, 2^2, 3^2, 4^2, ...`.
The explicit formula for the given sequence is `a_n = n^2`.
Substituting `n = 5`:
\(a_5 = 5^2\)
\(a_5 = 25\)
Therefore, the `5`th term of the sequence is `a_5 = 25`.
Example `4`: Determine the explicit formula that represents the geometric sequence `100, 50, 25, 12.5, \ldots`.
Solution:
Given geometric sequence: `100, 50, 25, 12.5, \ldots`
Here, the first term is `a_1 = 100` and the common ratio is `r = \frac{50}{100} = \frac{1}{2} `.
Explicit formula for a geometric sequence is:
`a_n = a_1 \times r^{(n-1)}`
Substituting the value of \(a_1 \) and \( r \), we get
`a_n = 100 \times \left(\frac{1}{2}\right)^{(n-1)}`
Example `5`: What is the common difference of the arithmetic sequence if its explicit formula is given by `a_n = 3n + 5`?
Solution:
Given explicit formula is `a_n = 3n + 5`.
Let us find the `1`st and `2`nd terms of the arithmetic sequence.
`a_1 = 3(1) + 5 = 8`
`a_2 = 3(2) + 5 = 11`
So, \(d = a_2 - a_1 = 11 - 8 = 3\)
So, the common difference of the arithmetic sequence is \( 3\).
Practice Problems
Q`1`. Determine the explicit formula that represents the arithmetic sequence `1, 3, 5, 7, 9, \ldots`.
- `a_n = 2n - 1`
- `a_n = 2n + 1`
- `a_n = 2n + 3`
- `a_n = 2n`
Answer: a
Q`2`. Determine the common ratio of the geometric sequence `16, 8, 4, 2, \ldots`.
- `\frac{1}{4}`
- `8`
- `\frac{1}{2}`
- `2`
Answer: c
Q`3`. What is the common difference of the arithmetic sequence if its explicit formula is given by `a_n = -2 + 4n`?
- `d = 2`
- `d = 4`
- `d = 6`
- `d = 8`
Answer: b
Q`4`. What is the common ratio of the geometric sequence if its explicit formula is given by `a_n = \frac{1}{2} \times 4^n`?
- `d = 16`
- `d = 12`
- `d = 8`
- `d = 4`
Answer: d
Q`5`. Determine the `5`th term of the geometric sequence where the first term `a_1 = 4` and the common ratio `r = 3`.
- `a_5 = 324`
- `a_5 = 355`
- `a_5 = 108`
- `a_5 = 210`
Answer: a
Frequently Asked Questions
Q`1`. What is an explicit formula?
Answer: An explicit formula is a mathematical expression that allows us to directly calculate any term in a sequence based on its position using a formula. It's usually denoted as `a_n` where `n` represents the position of the term in the sequence.
Q`2`. What is the explicit formula for an arithmetic sequence?
Answer: For an arithmetic sequence where each term increases or decreases by a constant difference `d`, the explicit formula is given by `a_n = a_1 + (n - 1)d`, where `a_1` is the first term of the sequence.
Q`3`. What is the explicit formula for a geometric sequence?
Answer: In a geometric sequence where each term is multiplied by a constant ratio `r` to get the next term, the explicit formula is represented as `a_n = a_1 \cdot r^{(n-1)}`, where `a_1` is the first term.
Q`4`. How to find the explicit formula for a sequence?
Answer: To find the explicit formula for a sequence, you need to identify the pattern that the terms follow. Once the pattern is recognized, you can express it using a mathematical formula, typically in terms of the position `n` in the sequence.
Q`5`. Are there any other types of sequences with explicit formulas?
Answer: Yes, apart from arithmetic, geometric, and harmonic sequences, there are various other types of sequences with explicit formulas, such as quadratic sequences, Fibonacci sequences, and polynomial sequences. Each type has its own distinct formula for generating terms based on their positions.