Problem 62 - Algebra and Trigonometry

Question

What is true about the sum of the exponents on \(a\) and \(b\) in any term in the expansion of \((a+b)^{n} ?\)

Step-by-Step Solution

Answer

The sum of the exponents of \(a\) and \(b\) in any term of an expansion of \((a+b)^{n}\) is always \(n\)

Step 1 - Consider an arbitrary term in the expansion

Consider any term in the expansion of \((a+b)^{n}\) can be of the form \(a^{k}b^{(n-k)}\), where \(k\) is an integer, \(0 ≤ k ≤ n\)

Step 2 - Sum of exponents

Sum the exponents of \(a\) and \(b\) in this arbitary term which is \(k + (n - k)\)

Step 3 - Simplify

Simplify \(k + (n - k)\). The \(k\) from \(n-k\) and \(k\) cancel out, yielding \(n\)

Step 4 - Conclusion

Hence, the sum of the exponents of \(a\) and \(b\) in any term of the expansion \((a+b)^{n}\) is \(n\)