1. **State the problem:** Find the first 4 terms, in ascending powers of $x$, of the binomial expansion of $ (2 - 5x)^8 $ and simplify each term. 2. **Formula and rules:** The binomial expansion for $(a + b)^n$ is given by: $$\sum_{k=0}^n \binom{n}{k} a^{n-k} b^k$$ where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$. 3. **Apply to our problem:** Here, $a = 2$, $b = -5x$, and $n = 8$. 4. **Calculate the first 4 terms ($k=0$ to $k=3$):** - Term 1 ($k=0$): $$\binom{8}{0} 2^{8} (-5x)^0 = 1 \times 256 \times 1 = 256$$ - Term 2 ($k=1$): $$\binom{8}{1} 2^{7} (-5x)^1 = 8 \times 128 \times (-5x) = -5120x$$ - Term 3 ($k=2$): $$\binom{8}{2} 2^{6} (-5x)^2 = 28 \times 64 \times 25x^2 = 44800x^2$$ - Term 4 ($k=3$): $$\binom{8}{3} 2^{5} (-5x)^3 = 56 \times 32 \times (-125x^3) = -2240000x^3$$ 5. **Write the first 4 terms in simplest form:** $$256 - 5120x + 44800x^2 - 2240000x^3$$ 6. **Part (b) - Value of $x$ to approximate $2.05^8$:** We write $2.05$ as $2 + 0.05$, so $x = -\frac{0.05}{5} = -0.01$ because the binomial is in terms of $(2 - 5x)$. **Final answers:** (a) The first 4 terms are: $$256 - 5120x + 44800x^2 - 2240000x^3$$ (b) The value of $x$ to use is: $$x = -0.01$$