l = a + (n-1)d \implies 996 = 108 + (n-1) \cdot 12

February 22, 2026

$l = a + (n-1)d \implies 996 = 108 + (n-1) \cdot 12$

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:

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.

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:

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

Step 3: Solve for n

Add 1 to both sides:

n = 74 + 1 n = 75

✅ Final Answer: There Are 75 Terms

So, 996 = 108 + (75 – 1) × 12 ✔ Verified!

Why This Formula Matters Beyond the Numbers

Understanding how to manipulate formulas like L = a + (n – 1)d helps in a variety of real-world applications:

Education: Calculating grade point averages in linear progression
Finance: Projecting interest growth or savings plans
Science: Modeling linear speed or temperature changes
Programming: Generating sequences or iterative outputs

Tips for Solving Such Equations

Always identify a, L, d, and n by reading the problem carefully.
Subtract the first term early to simplify the equation.
Divide by the common difference to isolate the term with n.
Always double-check your solution by plugging n back into the original formula.

Conclusion

Mastering the formula L = a + (n – 1)d unlocks powerful problem-solving abilities. Whether you’re tackling math homework, analyzing data trends, or designing systems with predictable growth, this simple yet versatile expression is a foundational tool.

Ready to solve your next sequence problem? Start by isolating (n – 1), divide by d, then add 1—just like in the 996 = 108 + (75 – 1) × 12 example!

Keywords: arithmetic progression, formula L = a + (n – 1)d, solve for n, step-by-step math examples, common difference, sequence mathematics, real-world applications of sequences, linear formulas, educational math tricks

Meta Description: Learn how to solve arithmetic progression problems using L = a + (n – 1)d with a practical example: 996 = 108 + (n – 1) × 12. Step-by-step guide to finding n.

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