Binomial Theorem(Part-5)

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Basanta Kumar Khanal

३२ असार २०८३

General Term of a Binomial Expansion

The general term is a formula that allows us to find any specific term in the expansion of a binomial expression without expanding the entire expression. It is one of the most useful applications of the Binomial Theorem, especially in competitive examinations and higher mathematics.

For the expansion of$$(a+b)^n,$$

the general term is$$T_{r+1}=\binom{n}{r}a^{\,n-r}b^{\,r}, \qquad r=0,1,2,\ldots,n.$$Here:

• $T_{r+1}$​ represents the $(r+1)^{\text{th}}$ term.
• $\binom{n}{r}$ is the binomial coefficient.
• The exponent of the first term $a$a decreases from $n$n to $0$0.
• The exponent of the second term $b$b increases from $0$0 to $n$n.

Example

Find the general term of $(2x+3)^8$(2x+3)8.

Using the formula,$$T_{r+1}=\binom{8}{r}(2x)^{8-r}(3)^r.$$This formula can be used to determine any required term, such as the 3rd, 5th, or last term, without expanding the complete expression.

Importance of the General Term

• Finds any specific term directly.
• Helps determine the coefficient of a particular power of a variable.
• Saves time by avoiding lengthy expansions.
• Widely used in algebra, calculus, probability, and competitive mathematics.

Conclusion

The general term is a powerful shortcut provided by the Binomial Theorem. By mastering this concept, students can solve complex expansion problems efficiently and develop a deeper understanding of algebraic expressions.