Chapter 5 – Binomial Theorem and Its Applications
1. Introduction to Binomial Theorem
The Binomial Theorem provides a systematic way to expand expressions of the form:
$(a + b)^n$, where $n$ is a non-negative integer.
This chapter is heavily used in:
- Algebraic expansions
- Coefficient problems
- Series and calculus (later chapters)
2. Factorial and Combination Recap
$n! = n \times (n-1) \times \dots \times 1$
$nC_r = \frac{n!}{r!(n-r)!}$
Binomial coefficients are combinations.
3. Statement of Binomial Theorem
$(a+b)^n = \sum_{r=0}^{n} nC_r \, a^{\,n-r} b^{\,r}$
This formula expands $(a+b)^n$ into $(n+1)$ terms.
4. General Term of the Expansion
General term:
$$T_{r+1} = nC_r \, a^{n-r} b^r$$
Most JEE problems are based on identifying or manipulating the general term.
5. Number of Terms
Total number of terms in $(a+b)^n = n+1$
6. Middle Term(s)
If $n$ is even → one middle term
If $n$ is odd → two middle terms
Middle term index:
- $\frac{n}{2}+1$ (for even $n$)
- $\frac{n+1}{2}$ and $\frac{n+3}{2}$ (for odd $n$)
7. Greatest Term in Binomial Expansion
The greatest term depends on the value of $\frac{b}{a}$.
Greatest term is $T_{r+1}$ where:
$$\frac{T_{r+1}}{T_r} \ge 1 \quad \text{and} \quad \frac{T_{r+2}}{T_{r+1}} \le 1$$
This topic is frequently asked in JEE Advanced.
8. Coefficient of a Particular Term
To find coefficient of $x^k$:
- Write general term
- Equate power of $x$
- Solve for $r$
9. Independent Term
Independent term → power of variable = 0
Very common in JEE Main.
10. Binomial Expansion with Fractional Terms
Expressions like:
- $(x + \frac{1}{x})^n$
- $(ax + \frac{b}{x})^n$
Power balancing is the key idea here.
11. Properties of Binomial Coefficients
$nC_r = nC_{n-r}$
$nC_r + nC_{r-1} = (n+1)C_r$
12. Special Values
- Sum of coefficients = $2^n$
- Alternating sum of coefficients = 0
13. Binomial Identities
$(1+1)^n = \sum nC_r = 2^n$
$(1-1)^n = \sum (-1)^r nC_r = 0$
14. Application in Approximation
Binomial theorem is used to approximate values like:
- $(1.01)^5$
- $(0.99)^4$
15. Common JEE Traps
- Wrong general term index
- Ignoring sign of terms
- Incorrect middle term selection
- Power mismatch in variables
16. Typical JEE Question Types
| Question Type | Approach |
|---|---|
| Coefficient of $x^k$ | Use general term |
| Middle term | Use $n$ parity |
| Greatest term | Use term ratio |
| Independent term | Balance powers |
17. Final Revision Checklist
You have mastered this chapter if you can:
- Write general term confidently
- Find coefficients without expanding fully
- Handle fractional powers
- Identify middle and greatest terms
- Use binomial identities efficiently