https://file.aiquickdraw.com/mathbot/file/1726181375590e66c1td8.jpg

To find the value of \( a \) in the expansion of \( (x + 1)^7 \), where there is a term of the form \( ax^3 \), we can use the binomial theorem. The binomial theorem gives the expansion of \( (x + 1)^7 \) as:

\[
(x + 1)^7 = \sum_{k=0}^{7} \binom{7}{k} x^k
\]

Here, \( \binom{7}{k} \) represents the binomial coefficient, which is calculated as:

\[
\binom{7}{k} = \frac{7!}{k!(7-k)!}
\]

We need to find the term where \( x^k = x^3 \). This occurs when \( k = 3 \).

The corresponding term in the expansion is given by:

\[
\binom{7}{3} x^3
\]

Now, we calculate \( \binom{7}{3} \):

\[
\binom{7}{3} = \frac{7!}{3!(7-3)!} = \frac{7 \times 6 \times 5}{3 \times 2 \times 1} = 35
\]

Thus, the term is \( 35x^3 \), which means \( a = 35 \).

### Answer:
\( a = 35 \).

Would you like further details or have any questions?

Here are 5 follow-up questions to explore further:

1. How does the binomial theorem apply to larger exponents, such as \( (x + 2)^{10} \)?
2. Can you find the general form for the \( x^2 \) term in \( (x + y)^7 \)?
3. How would the result change if the power was negative, like in \( (x + 1)^{-7} \)?
4. What is the relationship between binomial coefficients and Pascal's triangle?
5. How does the value of \( a \) change if the exponent changes, say from \( (x + 1)^7 \) to \( (x + 1)^8 \)?

**Tip:** When expanding binomials, identifying the desired term (like \( x^k \)) early can help speed up calculations!

The expansion of (x + 1)^7 has a term of the form ax^3. What is a?

Learn how to find the coefficient 'a' in the expansion of (x + 1)^7 for the term ax^3. This problem utilizes the binomial theorem and combinatorial methods, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation