1. **State the problem:** Expand $ (1 + x)^{10} $ up to the fifth term and use it to estimate the amount of 100000 invested at 5% compound interest for 10 years. 2. **Formula for binomial expansion:** $$(1 + x)^n = \sum_{k=0}^n \binom{n}{k} x^k = 1 + nx + \frac{n(n-1)}{2!}x^2 + \frac{n(n-1)(n-2)}{3!}x^3 + \frac{n(n-1)(n-2)(n-3)}{4!}x^4 + \cdots$$ 3. **Expand $ (1 + x)^{10} $ up to the fifth term:** Here, $n=10$ and $x$ is the interest rate as a decimal, $x=\frac{5}{100} = 0.05$. Calculate each term: - Term 1: $1$ - Term 2: $10x = 10 \times 0.05 = 0.5$ - Term 3: $\frac{10 \times 9}{2} x^2 = 45 \times (0.05)^2 = 45 \times 0.0025 = 0.1125$ - Term 4: $\frac{10 \times 9 \times 8}{6} x^3 = 120 \times (0.05)^3 = 120 \times 0.000125 = 0.015$ - Term 5: $\frac{10 \times 9 \times 8 \times 7}{24} x^4 = 210 \times (0.05)^4 = 210 \times 0.00000625 = 0.0013125$ Sum these terms: $$1 + 0.5 + 0.1125 + 0.015 + 0.0013125 = 1.6288125$$ 4. **Estimate the amount after 10 years:** Using compound interest formula: $$A = P(1 + \frac{r}{100})^n \approx P \times 1.6288125$$ Given $P = 100000$, $$A \approx 100000 \times 1.6288125 = 162881.25$$ 5. **Round to the nearest thousand:** $$162881.25 \approx 163000$$ **Final answer:** The amount after 10 years is approximately 163000.