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(vi). Total Number of Terms in the Expansion of (a+b)^(n+1) is
Introduction
In algebra, the binomial theorem is a fundamental concept that deals with the expansion of powers of a binomial expression. One of the most important aspects of the binomial theorem is the number of terms in the expansion of a power of a binomial. In this article, we will explore the total number of terms in the expansion of (a+b)^(n+1).
Binomial Theorem
The binomial theorem states that for any positive integer n,
$(a+b)^n = \sum_{k=0}^n {n \choose k} a^{n-k} b^k$
where ${n \choose k}$ is the binomial coefficient.
Expansion of (a+b)^(n+1)
To find the total number of terms in the expansion of (a+b)^(n+1), we can use the binomial theorem. The expansion of (a+b)^(n+1) is given by:
$(a+b)^{n+1} = \sum_{k=0}^{n+1} {{n+1} \choose k} a^{n+1-k} b^k$
Total Number of Terms
The total number of terms in the expansion of (a+b)^(n+1) is equal to the number of terms in the sum, which is n+2. Therefore, the total number of terms in the expansion of (a+b)^(n+1) is n+2.
Conclusion
In conclusion, the total number of terms in the expansion of (a+b)^(n+1) is n+2. This result is a direct consequence of the binomial theorem and is used extensively in many areas of mathematics, such as algebra, combinatorics, and probability.
