Explicit Formula - Algebra-1

1. Fundamental Concepts

The explicit formula of an arithmetic sequence is a formula that allows us to directly calculate any term \(a_n\) in the sequence without needing to know the previous terms. It is derived based on the first term \(a_1\) and the common difference d, where n represents the position of the term in the sequence (e.g., \(n=1\) for the first term, \(n=2\) for the second term, etc.).

2. Key Concepts

The general form of the explicit formula for an arithmetic sequence is:\(a_n = a_1 + d(n - 1)\) Here:
- \(a_n\) is the n-th term of the sequence,
- \(a_1\) is the first term,
- d is the common difference,
- n is the term number (a positive integer: \(n = 1, 2, 3, \dots\)).
The formula works because each term in an arithmetic sequence increases or decreases by the common difference d from the previous term. For the n-th term, the total change from the first term is \(d(n - 1)\) (since there are \(n - 1\) intervals between the first term and the n-th term).

3. Examples

Easy

Find the 5th term of an arithmetic sequence where \(a_1 = 3\) and \(d = 2\).

Solution: Using the explicit formula \(a_n = a_1 + d(n - 1)\): Substitute \(a_1 = 3\), \(d = 2\), and \(n = 5\):\(a_5 = 3 + 2(5 - 1) = 3 + 2(4) = 3 + 8 = 11\)

Medium

An arithmetic sequence starts with 10, and each subsequent term decreases by 4. What is the 12th term?

Solution: First, identify \(a_1 = 10\), \(d = -4\) (since the sequence decreases), and \(n = 12\). Using the explicit formula:\(a_{12} = 10 + (-4)(12 - 1) = 10 + (-4)(11) = 10 - 44 = -34\)

Hard

In an arithmetic sequence, the 3rd term is 15 and the 7th term is 31. Find the 20th term using the explicit formula.

Solution: Step 1: Use the explicit formula for the 3rd and 7th terms to set up equations.

For the 3rd term (\(n=3\)): \(a_3 = a_1 + d(3 - 1) \implies 15 = a_1 + 2d\)
For the 7th term (\(n=7\)): \(a_7 = a_1 + d(7 - 1) \implies 31 = a_1 + 6d\)

Step 2: Solve for \(a_1\) and d by subtracting the first equation from the second:\(31 - 15 = (a_1 + 6d) - (a_1 + 2d) \implies 16 = 4d \implies d = 4\)

Step 3: Substitute \(d = 4\) back into the first equation to find \(a_1\):\(15 = a_1 + 2(4) \implies 15 = a_1 + 8 \implies a_1 = 7\)

Step 4: Find the 20th term using \(a_1 = 7\), \(d = 4\), and \(n = 20\):\(a_{20} = 7 + 4(20 - 1) = 7 + 4(19) = 7 + 76 = 83\)

4. Problem-Solving Techniques

Identify known values: First, determine what is given (\(a_1\), d, a specific term \(a_n\), or n) and what needs to be found.
Plug into the formula: Substitute the known values into the explicit formula \(a_n = a_1 + d(n - 1)\) and solve for the unknown variable.
Check consistency: If solving for d or \(a_1\) using two known terms, verify the result by plugging it back into the formula to ensure it produces the correct terms.
Handle negative differences: Remember that a negative d indicates a decreasing sequence, and a positive d indicates an increasing sequence.