Binomial Theorem Cheatsheet
Cheat Codes & Shortcuts
- Binomial Theorem: \( (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \), where \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
- Binomial Coefficient: \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \), counts combinations.
- Pascal’s Triangle: Row \( n \) gives coefficients for \( (a + b)^n \).
- General Term: In \( (a + b)^n \), the \( (k+1) \)-th term is \( T_{k+1} = \binom{n}{k} a^{n-k} b^k \).
- Sum of Coefficients: For \( (1 + 1)^n \), sum is \( 2^n \).
- Binomial Expansion for Non-Integer \( n \): \( (1 + x)^n = 1 + nx + \frac{n(n-1)}{2!} x^2 + \cdots \), valid for \( |x| < 1 \).
- Number of Terms: For \( (a + b)^n \), there are \( n+1 \) terms.
- Middle Term: For even \( n \), middle term at \( k = \frac{n}{2} \); for odd \( n \), two middle terms at \( k = \frac{n-1}{2}, \frac{n+1}{2} \).
- Application: Approximate \( (1 + x)^n \) for small \( x \).
- Combinatorial Identity: \( \sum_{k=0}^n \binom{n}{k} = 2^n \).
Quick Reference Table
| Type | Form | Solution/Method |
|---|---|---|
| Binomial Expansion | \( (x + 2)^3 \) | \( \binom{3}{0} x^3 2^0 + \binom{3}{1} x^2 2^1 + \binom{3}{2} x 2^2 + \binom{3}{3} 2^3 \) |
| General Term | \( (x + y)^5 \) | \( T_{k+1} = \binom{5}{k} x^{5-k} y^k \) |
| Coefficient Sum | \( (1 + 1)^4 \) | \( 2^4 = 16 \) |
| Non-Integer Expansion | \( (1 + x)^{1/2} \) | \( 1 + \frac{x}{2} - \frac{x^2}{8} + \cdots \), for \( |x| < 1 \) |
| Middle Term | \( (x + y)^4 \) | At \( k = 2 \), term is \( \binom{4}{2} x^2 y^2 = 6x^2 y^2 \) |
| Application | Approximate \( (1.02)^5 \) | Use \( (1 + 0.02)^5 \approx 1 + 5(0.02) + \frac{5 \cdot 4}{2} (0.02)^2 \) |
Advice
First Step: Identify \( n \), \( a \), and \( b \) in \( (a + b)^n \).
Coefficients: Use Pascal’s Triangle or compute \( \binom{n}{k} \).
General Term: Use \( T_{k+1} = \binom{n}{k} a^{n-k} b^k \) for specific terms.
Approximations: For small \( x \), use first few terms of expansion.
Verify: Check sum of coefficients or expand small cases manually.
Binomial Theorem Quick Tips
- Binomial Formula: \( (a + b)^n = \sum_{k=0}^n \binom{n}{k} a^{n-k} b^k \).
- Pascal’s Triangle: Quick way to find coefficients for small \( n \).
- Middle Term: For even \( n \), use \( k = \frac{n}{2} \); for odd \( n \), check both middle terms.
- Non-Integer \( n \): Use \( (1 + x)^n \approx 1 + nx \) for small \( x \).
- Sum of Coefficients: Set \( a = b = 1 \) to get \( 2^n \).
Binomial Theorem Speed Quiz
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