Explicit Form Calculator – Loan calculator

Explicit Form Sequence Calculator

First Term (a₁):

Term Number (n) to find:

Sequence Type:
Arithmetic Geometric

Common Difference (d):

Common Ratio (r):

Calculation Results:

" + "Explicit Formula: " + explicitFormulaString + "" + "The " + n + "^th Term (a_{" + n + "}): " + nthTerm.toFixed(4) + "" + "Sum of the First " + n + " Terms (S_{" + n + "}): " + sumOfNTerms.toFixed(4) + ""; } // Initialize the display based on default radio button selection toggleCommonInput();

Understanding Explicit Form Sequences

An explicit form, or explicit formula, for a sequence is a powerful mathematical tool that allows you to directly calculate any term in a sequence without needing to know the preceding terms. Instead of defining a term based on the one before it (which is a recursive definition), an explicit formula defines the n^th term as a function of its position 'n'. This calculator helps you find specific terms and sums for two common types of sequences: arithmetic and geometric.

Arithmetic Sequences

An arithmetic sequence is a sequence of numbers such that the difference between consecutive terms is constant. This constant difference is called the common difference (d).

Explicit Formula for the n^th term (a_n):
a_n = a₁ + (n - 1)d
Where:
- a_n is the n^th term
- a₁ is the first term
- n is the term number
- d is the common difference
Sum of the first n terms (S_n):
S_n = n/2 * (2a₁ + (n - 1)d) or S_n = n/2 * (a₁ + a_n)

Arithmetic Sequence Example:

Consider an arithmetic sequence where the first term (a₁) is 3 and the common difference (d) is 4. We want to find the 10^th term (a₁₀) and the sum of the first 10 terms (S₁₀).

Inputs: First Term (a₁) = 3, Term Number (n) = 10, Common Difference (d) = 4.
Explicit Formula: a_n = 3 + (n – 1) * 4
10^th Term (a₁₀):
a₁₀ = 3 + (10 – 1) * 4 = 3 + 9 * 4 = 3 + 36 = 39
Sum of the First 10 Terms (S₁₀):
S₁₀ = 10/2 * (2*3 + (10 – 1)*4) = 5 * (6 + 9*4) = 5 * (6 + 36) = 5 * 42 = 210

Geometric Sequences

A geometric sequence is a sequence of numbers where each term after the first is found by multiplying the previous one by a fixed, non-zero number called the common ratio (r).

Explicit Formula for the n^th term (a_n):
a_n = a₁ * r^{(n - 1)}
Where:
- a_n is the n^th term
- a₁ is the first term
- n is the term number
- r is the common ratio
Sum of the first n terms (S_n):
S_n = a₁ * (1 - rⁿ) / (1 - r) (if r ≠ 1)
S_n = n * a₁ (if r = 1)

Geometric Sequence Example:

Consider a geometric sequence where the first term (a₁) is 2 and the common ratio (r) is 3. We want to find the 5^th term (a₅) and the sum of the first 5 terms (S₅).

Inputs: First Term (a₁) = 2, Term Number (n) = 5, Common Ratio (r) = 3.
Explicit Formula: a_n = 2 * 3^{(n – 1)}
5^th Term (a₅):
a₅ = 2 * 3^{(5 – 1)} = 2 * 3⁴ = 2 * 81 = 162
Sum of the First 5 Terms (S₅):
S₅ = 2 * (1 – 3⁵) / (1 – 3) = 2 * (1 – 243) / (-2) = 2 * (-242) / (-2) = 242

How to Use the Calculator

Enter the First Term (a₁): This is the starting value of your sequence.
Enter the Term Number (n): This is the position of the term you want to find (e.g., 5 for the 5^th term).
Select Sequence Type: Choose whether your sequence is Arithmetic or Geometric.
Enter Common Difference (d) or Common Ratio (r):
- If Arithmetic, enter the constant difference between terms.
- If Geometric, enter the constant multiplier between terms.
Click "Calculate": The calculator will display the explicit formula for the sequence, the value of the n^th term, and the sum of the first n terms.

This tool simplifies the process of working with sequences, allowing you to quickly derive terms and sums for various mathematical and real-world applications.