Let's address both parts of the question from the image you've uploaded.

Part (a):

You are asked to find the first four terms of the binomial expansion of $\left(2 + \frac{x}{5}\right)^{10}$.

The binomial expansion formula is given by: $(a + b)^n = \sum_{k=0}^{n} \binom{n}{k} a^{n-k} b^k$ Here, $a = 2$, $b = \frac{x}{5}$, and $n = 10$.

Now, we will expand the first four terms:

1. First term: $k = 0$ $\binom{10}{0} (2)^{10} \left(\frac{x}{5}\right)^0 = 1 \cdot 2^{10} = 1024$

2. Second term: $k = 1$ $\binom{10}{1} (2)^9 \left(\frac{x}{5}\right)^1 = 10 \cdot 2^9 \cdot \frac{x}{5} = 10 \cdot 512 \cdot \frac{x}{5} = 1024 \cdot \frac{x}{5} = 204.8x$

3. Third term: $k = 2$ $\binom{10}{2} (2)^8 \left(\frac{x}{5}\right)^2 = 45 \cdot 256 \cdot \left(\frac{x^2}{25}\right) = 11520 \cdot \frac{x^2}{25} = 460.8x^2$

4. Fourth term: $k = 3$ $\binom{10}{3} (2)^7 \left(\frac{x}{5}\right)^3 = 120 \cdot 128 \cdot \left(\frac{x^3}{125}\right) = 15360 \cdot \frac{x^3}{125} = 122.88x^3$

Thus, the first four terms of the expansion are: $1024 + 204.8x + 460.8x^2 + 122.88x^3$

Part (b):

You are asked to substitute an appropriate value for $x$ and find an approximate value for $2.1^{10}$.

We notice that $2.1 = 2 + 0.1$, so let $x = 0.5$ (since $\frac{x}{5} = 0.1$).

Now, substitute $x = 0.5$ into the binomial expansion: $1024 + 204.8(0.5) + 460.8(0.5)^2 + 122.88(0.5)^3$

Let's calculate this step-by-step:

1. $204.8 \times 0.5 = 102.4$
2. $460.8 \times (0.5)^2 = 460.8 \times 0.25 = 115.2$
3. $122.88 \times (0.5)^3 = 122.88 \times 0.125 = 15.36$

Now, adding all the terms together: $1024 + 102.4 + 115.2 + 15.36 = 1256.96$

So, an approximate value for $2.1^{10}$ is 1256.96.

Would you like more details or have any other questions?

Here are 5 related questions to consider:

1. How does the binomial expansion work for negative exponents?
2. Can you explain why we use binomial coefficients in the expansion?
3. What would the next term in the expansion be for this problem?
4. How can we approximate values for non-integer exponents using series expansions?
5. How accurate is the approximation compared to the exact value of $2.1^{10}$?

Tip: Binomial expansion can also be used for approximate calculations when the second term is small, as seen in part (b).