l = a + (n-1)d \implies 996 = 108 + (n-1) \cdot 12 - Redraw
Understanding the Formula: How to Solve for n Using L = a + (n – 1)d
With Step-by-Step Example: 996 = 108 + (n – 1) × 12
Understanding the Formula: How to Solve for n Using L = a + (n – 1)d
With Step-by-Step Example: 996 = 108 + (n – 1) × 12
When it comes to arithmetic progressions, one of the most useful formulas is:
Understanding the Context
L = a + (n – 1)d
Where:
- L = the last term in the sequence
- a = the first term
- n = the number of terms
- d = the common difference between successive terms
This formula is essential for solving word problems involving sequences, financial calculations, or mathematical progressions. In this article, we’ll break down how to use this formula, walk through a practical example—such as solving 996 = 108 + (n – 1) × 12—and explain how you can apply it to similar problems.
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Key Insights
What Does the Formula Mean?
The formula L = a + (n – 1)d describes how each term in an arithmetic sequence progresses. Starting from the first term a, each subsequent term increases (or decreases) by a constant d. The expression (n – 1)d calculates the total increment over n – 1 steps.
Applying the Formula: Step-By-Step Example
Let’s solve the equation step-by-step:
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Given:
996 = 108 + (n – 1) × 12
This represents an arithmetic sequence where:
- L = 996 (the final term)
- a = 108 (the first term)
- d = 12 (the common difference)
- n = the unknown (number of terms)
Step 1: Isolate the term containing n
Subtract the first term from both sides:
996 – 108 = (n – 1) × 12
888 = (n – 1) × 12
Step 2: Divide both sides by 12
888 ÷ 12 = n – 1
74 = n – 1
