Binomial Theorem - Binomial theorem for positive integral indices

Definition of Binomial Theorem: The Binomial Theorem for positive integral indices states that for any positive integer $n$ and any real numbers $a$ and $b$, the expansion of $(a+b)^n$ consists of $(n+1)$ terms. The coefficients of these terms are binomial coefficients denoted by $^nC_r$.

Expansion Pattern and Sum of Indices: In the expansion of $(a+b)^n$, the power of '$a$' starts at $n$ and decreases by 1 in each subsequent term until it reaches 0. Simultaneously, the power of '$b$' starts at 0 and increases by 1 until it reaches $n$. Visually, if you look at any term, the sum of the exponents of $a$ and $b$ is always equal to the original index $n$.

Pascal's Triangle: The coefficients of the binomial expansion can be arranged in a geometric, triangular pattern called Pascal's Triangle. It begins with 1 at the top ($n=0$). Each row starts and ends with 1, and every interior number is the sum of the two numbers directly above it. This provides a clear visual symmetry to the coefficients of the expansion.

The General Term ($T_{r+1}$): Any specific term in the expansion can be located using the general term formula $T_{r+1} = ^nC_r a^{n-r} b^r$. Here, $r$ represents the index of the combination and is always one less than the position of the term (e.g., for the 5th term, $r=4$).

Middle Terms: The number of terms in the expansion is $(n+1)$. If $n$ is even, there is a single middle term at position $(\frac{n}{2} + 1)$. If $n$ is odd, the number of terms $(n+1)$ is even, meaning there are two middle terms at positions $(\frac{n+1}{2})$ and $(\frac{n+1}{2} + 1)$. This can be visualized as the 'peak' of the distribution of coefficients.

Symmetry of Coefficients: The coefficients in the expansion are symmetrical. The first coefficient $^nC_0$ is equal to the last $^nC_n$, the second $^nC_1$ is equal to the second-to-last $^nC_{n-1}$, and so on. This follows the visual symmetry of Pascal's Triangle and the mathematical identity $^nC_r = ^nC_{n-r}$.