Explicit Linear Growth Formula Explained
What Is the Explicit Linear Growth Formula?
The explicit linear growth formula lets you find any term in a sequence without listing all the previous ones. That's it. That's the whole point.
Most students first encounter this in arithmetic sequences. You have a pattern where each term increases by a fixed amount. Instead of writing out every single term to find the 50th one, you plug numbers into a single equation.
This formula is also called the explicit formula for arithmetic sequences. Some textbooks call it the "nth term formula." Same thing.
The Formula
Here's the explicit linear growth formula:
an = a1 + (n - 1)d
Where:
- an = the term you want to find
- a1 = the first term in the sequence
- n = which term number you need
- d = the common difference (what you add each time)
Breaking Down Each Component
The First Term (a₁)
This is simply where your sequence starts. Look at the problem and identify the very first number given. That's your a₁. No tricks here.
The Common Difference (d)
Take any term and subtract the term before it. The result is your common difference. If the sequence is 3, 7, 11, 15..., then d = 7 - 3 = 4. The difference stays constant throughout a true arithmetic sequence.
The Term Number (n)
This is just the position. The 1st term is n=1. The 20th term is n=20. Make sure you're clear on which term you need before plugging in.
How to Use It: Step-by-Step
Let's work through a real example.
Problem: Find the 15th term of the sequence 5, 9, 13, 17...
Step 1: Identify a₁
The first term is 5. So a₁ = 5.
Step 2: Find d
9 - 5 = 4. The common difference is 4. So d = 4.
Step 3: Identify n
We need the 15th term. So n = 15.
Step 4: Plug into the formula
a₁₅ = 5 + (15 - 1) × 4
a₁₅ = 5 + 14 × 4
a₁₅ = 5 + 56
a₁₅ = 61
That's it. Four steps. No guessing, no writing out 15 terms.
Explicit vs. Recursive: What's the Difference?
There are two ways to define a sequence. You need to know which one you're using.
| Feature | Explicit Formula | Recursive Formula |
|---|---|---|
| Finds any term directly | ✅ Yes | ❌ No — requires previous terms |
| Needs the first term | Once, at the start | Every single step |
| Best for finding distant terms | ✅ Efficient | ❌ Slow (must calculate all terms before) |
| Best for pattern recognition | ❌ Less intuitive | ✅ Shows the pattern clearly |
The recursive formula looks like this: an = an-1 + d, with a₁ specified. To find the 100th term recursively, you'd need to calculate terms 1 through 99 first. The explicit formula skips all that.
Common Mistakes That Will Cost You Points
- Forgetting the (n-1). The formula is a₁ + (n-1)d, not a₁ + nd. The subtraction is there for a reason. Without it, you're off by one term.
- Getting a₁ wrong. Double-check that you're using the first term, not the second. Easy to mess up when scanning a problem.
- Sign errors with negative d. If the sequence decreases (like 20, 15, 10...), your d is negative. That's fine. Just keep track of the negative sign through the calculation.
- Confusing n with an. n is the position. an is the value at that position. Don't mix them up.
Quick Reference Table
| Given Information | Formula to Use |
|---|---|
| Find term when given a₁ and d | an = a₁ + (n-1)d |
| Find d when given two terms | d = (an - am) / (n - m) |
| Find a₁ when given an and d | a₁ = an - (n-1)d |
Practice Problem
Try this one before scrolling:
Find the 25th term of: 100, 93, 86, 79...
Solution:
- a₁ = 100
- d = 93 - 100 = -7
- n = 25
- a₂₅ = 100 + (25 - 1)(-7)
- a₂₅ = 100 + 24(-7)
- a₂₅ = 100 - 168
- a₂₅ = -68
When You'll Actually Use This
Linear growth formulas show up in:
- Salary increases with fixed annual raises
- Depreciation calculations
- Loan amortization (some parts)
- Physics problems with constant acceleration
- Any situation where something changes by the same amount each period
The math is straightforward. Identify your values, plug them in, solve. The hard part is reading the problem correctly and not rushing through the setup.