Binomial Theorem - General and Middle Term in Binomial Expansion

The General Term of a binomial expansion $(a+b)^n$ is denoted by $T_{r+1}$. It represents any arbitrary term in the sequence of the expansion, where $r$ ranges from $0$ to $n$. Visually, as $r$ increases, the power of the first term $a$ decreases from $n$ to $0$ while the power of the second term $b$ increases from $0$ to $n$.

In the expansion of $(a+b)^n$, the total number of terms is always $n+1$. This is because the index $r$ in the general term starts at $0$ and ends at $n$. For example, a quadratic expansion $(a+b)^2$ results in $3$ terms, which can be visualized as three points on a horizontal axis.

The General Term formula $T_{r+1} = \,^nC_r a^{n-r} b^r$ is used to find a specific term (like the 5th or 10th term) without expanding the entire expression. If the binomial is $(a-b)^n$, the formula incorporates the negative sign as $T_{r+1} = \,^nC_r a^{n-r} (-b)^r$.

When $n$ is an even number, the expansion $(a+b)^n$ has an odd number of terms $(n+1)$. Consequently, there is exactly one middle term. This term is the $(\frac{n}{2} + 1)$-th term. Visually, the coefficients in this expansion form a symmetric curve (like a bell shape) with a single peak at the center.

When $n$ is an odd number, the expansion $(a+b)^n$ has an even number of terms $(n+1)$. In this case, there are two middle terms. These are the $(\frac{n+1}{2})$-th term and the $(\frac{n+3}{2})$-th term. The coefficients of these two middle terms are equal due to the property $\,^nC_r = \,^nC_{n-r}$, appearing as two equal peaks at the center of the distribution.

The 'Term Independent of $x$' is a specific term in an expansion involving $x$ where the net exponent of $x$ is zero. To find this, we express the general term, collect all powers of $x$, and set the sum of these powers to $0$. This term represents the constant value where the variable $x$ disappears from the algebraic expression.

Binomial coefficients $\,^nC_r$ exhibit symmetry. The coefficient of the $k$-th term from the beginning is the same as the coefficient of the $k$-th term from the end. This is visually represented in Pascal's Triangle, where each row is a palindrome of numbers.