Binomial Theorem - Simple Applications

The Binomial Theorem provides a formula for the expansion of $(a+b)^n$ for any positive integer $n$. Visually, this expansion follows the structure of Pascal's Triangle, a symmetric pyramid where each number is the sum of the two directly above it, representing the binomial coefficients ${^nC_r}$.

The total number of terms in the expansion of $(a+b)^n$ is always $n+1$. For example, a squared binomial $(a+b)^2$ has 3 terms, and a cubed binomial $(a+b)^3$ has 4 terms.

The General Term, denoted as $T_{r+1}$, is a single expression that represents any term in the expansion. It is given by $T_{r+1} = {^nC_r} a^{n-r} b^r$. This formula is the primary tool for finding a specific power of $x$ or a term at a specific position without writing the full expansion.

The Middle Term depends on the value of $n$. If $n$ is even, the expansion has an odd number of terms, resulting in one middle term at position $(\frac{n}{2} + 1)$. If $n$ is odd, the expansion has an even number of terms, resulting in two middle terms at positions $(\frac{n+1}{2})$ and $(\frac{n+3}{2})$. Visually, these terms correspond to the peak values in the distribution of coefficients.

The term independent of $x$ (the constant term) occurs when the total exponent of $x$ in the general term equals zero. To find it, one must aggregate all powers of $x$ from both parts of the binomial and solve for $r$ such that the resulting exponent is 0.

The sum of all binomial coefficients in the expansion of $(1+x)^n$ is $2^n$. This is derived by substituting $x=1$ into the identity. Visually, this is equivalent to summing all the values in the $n^{th}$ row of Pascal's Triangle.

Binomial expansions are used for numerical approximations. For example, $(1.01)^{10}$ can be written as $(1 + 0.01)^{10}$. By calculating only the first few terms of the expansion, one can obtain a highly accurate approximation because subsequent terms involve powers of $0.01$ which become insignificantly small.