Binomial Theorem

Class 11 & 12 Essentials

1. The Expansion Formula

For any positive integer n:

$$ (a+b)^n = \sum_{r=0}^{n} {}^nC_r \cdot a^{n-r} \cdot b^r $$

$$ = {}^nC_0 a^n + {}^nC_1 a^{n-1}b + \dots + {}^nC_n b^n $$

Important Observation:

  • Total number of terms = $$ n + 1 $$
  • Sum of powers of ‘a’ and ‘b’ in every term is always $$ n $$.

2. Finding Specific Terms

General Term ($$ T_{r+1} $$)

Used to find the $$ (r+1)^{th} $$ term:

$$ T_{r+1} = {}^nC_r \cdot a^{n-r} \cdot b^r $$

Middle Term(s)

  • If n is Even: One middle term
    $$ (\frac{n}{2} + 1)^{th} $$ term.
  • If n is Odd: Two middle terms
    $$ (\frac{n+1}{2})^{th} $$ and $$ (\frac{n+1}{2} + 1)^{th} $$ term.

3. Properties & Approximations

Property Formula
Sum of all Coefficients $$ {}^nC_0 + {}^nC_1 + \dots + {}^nC_n = 2^n $$
Sum of Odd/Even Coeffs $$ C_0 + C_2 + \dots = C_1 + C_3 + \dots = 2^{n-1} $$
Symmetry $$ {}^nC_r = {}^nC_{n-r} $$

Binomial Approximation (Physics Tool):

If $$ n $$ is any rational number (negative/fraction) and $$ |x| << 1 $$:

$$ (1+x)^n \approx 1 + nx $$

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