Annual growth rate from a truncated binomial expansion

A town has a population of \(50\,000\) at the start of 2026. A planner assumes the population grows by the same percentage, \(p\%\), every year, so that its population \(n\) years later is \(50\,000\left(1+\dfrac{p}{100}\right)^{n}\).

1. Write down the first three terms in the expansion of \((1+x)^{10}\) in ascending powers of \(x\).[2]
2. The planner wants the population at the start of 2036 to be \(60\,000\). Taking \(x=\dfrac{p}{100}\) and ignoring terms in \(x^3\) and higher powers of \(x\), form a quadratic equation in \(x\) and hence estimate \(p\), correct to 2 decimal places.[4]
3. Explain why ignoring the terms in \(x^3\) and higher powers of \(x\) makes your estimate in (b) slightly too large rather than too small.[2]
4. Using your estimate of \(p\), calculate the population the model predicts for the start of 2036, to the nearest hundred, and comment on whether the estimate is good enough for the planner.[2]