Question 5 Consider the expression \( (1+10 x)^{-\frac{1}{2}} \) a) Write out the binomial expansion up to and including the \( x^{2} \) term. \( \square \)

The binomial expansion of \( (1+10x)^{-\frac{1}{2}} \) can be expressed using the binomial series formula. The first few terms are: \[ (1 + 10x)^{-\frac{1}{2}} = 1 - \frac{1}{2}(10x) + \frac{(-\frac{1}{2})(-\frac{3}{2})}{2!}(10x)^2 + \ldots \] Calculating the terms, we get: \[ 1 - 5x + \frac{15}{4} \cdot 100x^2 = 1 - 5x + \frac{375}{4}x^2 \] Thus, the expansion up to \( x^2 \) is: \[ 1 - 5x + \frac{375}{4}x^2 \] --- Did you know that the binomial expansion was first systematically studied by Sir Isaac Newton? He introduced it in the 17th century, extending Pascal’s triangle concepts to non-integer and even fractional exponents. It’s fascinating how his discoveries birthed a fundamental tool that we still use today in fields ranging from algebra to statistics! When using binomial expansion, one common mistake is forgetting the factorial in the denominator for higher-order terms. It’s crucial to apply \(\frac{n!}{k!(n-k)!}\) properly, especially in complex expansions. A good tip is to write out the first few terms of the series to catch errors—nothing like double-checking our math to keep those mistakes at bay!