Find the first five terms, in ascending powers of 𝑥, in the expansion of (1 + 𝑥)^7


To find the first five terms in the expansion of \((1 + x)^7\), we can use the **Binomial Theorem**, which states that:

\[
(1 + x)^n = \sum_{k=0}^{n} \binom{n}{k} x^k
\]

Where:
- \( \binom{n}{k} \) is the binomial coefficient, given by \( \binom{n}{k} = \frac{n!}{k!(n-k)!} \).
- \( n = 7 \) in this case.

We'll calculate the first five terms (i.e., up to \(x^4\)).

1. **For \(k = 0\)**:
   \[
   \binom{7}{0} x^0 = 1
   \]
   
2. **For \(k = 1\)**:
   \[
   \binom{7}{1} x^1 = 7x
   \]
   
3. **For \(k = 2\)**:
   \[
   \binom{7}{2} x^2 = \frac{7 \times 6}{2} x^2 = 21x^2
   \]
   
4. **For \(k = 3\)**:
   \[
   \binom{7}{3} x^3 = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} x^3 = 35x^3
   \]
   
5. **For \(k = 4\)**:
   \[
   \binom{7}{4} x^4 = \frac{7 \times 6 \times 5 \times 4}{4 \times 3 \times 2 \times 1} x^4 = 35x^4
   \]

Thus, the first five terms in the expansion of \((1 + x)^7\) are:

\[
1 + 7x + 21x^2 + 35x^3 + 35x^4
\]

Would you like more details or have any questions? Here are five related questions to explore further:

1. How is the binomial coefficient calculated in general?
2. What is the full expansion of \((1 + x)^7\)?
3. What would the expansion look like for negative exponents, such as \((1 + x)^{-7}\)?
4. How can the binomial theorem be applied to higher-order powers, like \((1 + x)^{10}\)?
5. What is the general formula for the expansion of \((a + b)^n\)?

**Tip:** The binomial theorem is a powerful tool for expanding expressions of the form \((1 + x)^n\), and it is used extensively in algebra and calculus for approximation techniques!

Find the first five terms, in ascending powers of 𝑥, in the expansion of (1 + 𝑥)^7

Explore how to find the first five terms in the expansion of (1 + x)^7 using the binomial theorem. This solution covers algebraic concepts such as binomial coefficients and polynomial expansions. Ideal for students in Grades 10-12.

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