Explicit Formula Calculator
Use this Explicit Formula Calculator to determine the nth term for any arithmetic or geometric sequence. Just input the first term, common difference or ratio, and the term position (n) to instantly find the specific term.
What is an Explicit Formula?
Explicit Formula is a mathematical expression used to find any term in a sequence directly without referring to previous terms. It expresses the nth term of a sequence as a function of its position (n).
Explicit Formula:
The equation of an explicit formula for an arithmetic sequence is:
aₙ = a₁ + (n−1) ⋅ d
For geometric sequences, the explicit formula is:
aₙ = a₁ . r⁽ⁿ⁻¹⁾
How to Solve Explicit Sequences?
- Identify the type of sequence
- Find the first term (a1)
- Find the common difference (d) or common ratio (r)
- Determine the value of (n) term
- Write the explicit formula and put values to calculate the answer
Example:
The sequence 3, 6, 9, 12,... follows an arithmetic pattern. Determine the explicit formula for the nᵗʰ term of this sequence. Then, use the formula to find the 8ᵗʰ term of the sequence.
Given:
- First term ‘a₁’ = 3
- Common difference ‘d’ = 3
- Term number to Find ‘n’ = 8
Solution:
Step 1: Identify the type of sequence
Sequence Type: Arithmetic
Formula for the arithmetic sequences:
aₙ = a₁ + (n−1) ⋅ d
Step 2: Plug in the Known Values
a₈ = 3 + (8 - 1) . 3
Step 3: Simplify
Subtract inside the parentheses:
a₈ = 3 + 7 . 3
a₈ = 3 + 21
a₈ = 24
Step 4: Final Answer
✅ So, the 8th term of the sequence is 24.
Instead of solving the problem manually, there is another solution to get an instant and accurate solution. Whether it's an arithmetic or geometric sequence, simply enter a few values into the explicit formula calculator and get the exact term you're looking for in seconds.
How Does the Explicit Formula Calculator Work?
Follow these simple steps:
- Choose Sequence Type: Select the sequence type, either arithmetic or geometric
- Enter First Term (a₁): Input the first number of the sequence that is indicated as a₁
- Enter Common Difference (d) or Ratio (r): Provide the value that defines how the sequence progresses
- Enter Term Number (n): Specify which term (e.g., 5th, 10th) you want to calculate
- Click Button: Press the button “CALCULATE” to view the result based on your inputs instantly
Importance of Explicit Formula Calculator:
- Efficiency for Large Sequences: Saves time and effort when dealing with long sequences or high-term positions
- Accurate Results: Our tool helps to reduce human errors in manual calculations and provides you with step-by-step answers
- User-Friendly Interface: This tool provides you with a simple interface with easy input fields and provides an instant solution
- Supports Learning: Our tool is an ideal choice for you because it helps students and teachers to understand the sequences easily.
- Compatible with all Devices: This tool is fully responsive and accessible on any device, whether it's a mobile, tablet, or desktop.
FAQs:
When should I use an explicit formula instead of a recursive one?
When you need to know a specific term directly, use an explicit formula instead of calculating all previous terms.
What is the explicit formula for 2, 4, 6, 8?
This is an example of the arithmetic sequence where:
- First term a1 = 2
- Common Difference d = 2
The explicit formula is: aₙ = 2 + (n -1) x 2
Simplified:
aₙ = 2n
So, the explicit formula for this sequence is aₙ = 2n.
What is explicit formula for arithmetic sequence 4, 5, 29, 3, 0, 17, 15, 13, 10?
This sequence is not arithmetic because the differences between terms are not consistent:
- 5 − 4 = 1
- 29 − 5 = 24
- 30 − 29 = 1
- 17 − 30 = 13
- 15 − 17 = 2
- 13 − 15 = 2
- 10 − 13 = 3
Since the difference between consecutive terms is not constant, this is not an arithmetic sequence, and therefore, no explicit arithmetic formula exists for it.
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