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Let's work through writing each expression in sigma notation:

### (a) \( (x + 5) - (x + 3)^2 + (x + 1)^3 - (x - 1)^4 \)

This expression looks like it could follow a pattern, where each term is of the form \( (x + n)^k \), with \( n \) decreasing by 2 and \( k \) increasing by 1. We can rewrite it using sigma notation:

\[
\sum_{k=1}^{4} (-1)^{k+1} (x + (7 - 2k))^k
\]

Explanation:
- The exponent \( k \) increases from 1 to 4.
- The expression \( 7 - 2k \) adjusts the numbers inside the parentheses: for \( k = 1 \), it's 5, for \( k = 2 \), it's 3, and so on.
- The alternating signs are handled by \( (-1)^{k+1} \).

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### (b) \( \frac{1}{3^3} + \frac{2^2}{4^3} + \frac{3^2}{5^3} + \dots + \frac{10^2}{11^3} \)

The general term seems to be \( \frac{n^2}{(n+2)^3} \), where \( n \) runs from 1 to 10. Thus, in sigma notation, this is:

\[
\sum_{n=1}^{10} \frac{n^2}{(n+2)^3}
\]

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### (c) \( a_3 + a_6 + a_9 + \dots + a_{81} \)

This is a sum of terms \( a_{3n} \), where \( n \) starts at 1 and increases until \( 3n = 81 \), i.e., \( n \) goes from 1 to 27. So the sigma notation for this is:

\[
\sum_{n=1}^{27} a_{3n}
\]

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Do you want further details on any of the steps, or clarification on any part of the process?

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### Additional Practice:
1. How would you expand \( \sum_{k=1}^{4} (-1)^{k+1} (x + (7 - 2k))^k \)?
2. Can you express \( \sum_{n=1}^{10} \frac{n^2}{(n+2)^3} \) in expanded form?
3. What happens if you extend the sum \( a_3 + a_6 + a_9 + \dots \) beyond \( a_{81} \)?
4. How can sigma notation simplify complex series?
5. How does alternating the sign in a sum impact its overall value?

### Tip:
When identifying a series for sigma notation, look for consistent patterns in both the base of the terms and the exponents. This makes it easier to write a general formula for the sum.

Write the following in sigma notation: (a) (x + 5) - (x + 3)^2 + (x + 1)^3 - (x - 1)^4, (b) 1/3^3 + 2^2/4^3 + 3^2/5^3 + ... + 10^2/11^3, (c) a_3 + a_6 + a_9 + ... + a_81

Learn how to express mathematical sequences and series in sigma notation. This example covers powers of binomials and sequences with specific patterns, providing a detailed step-by-step solution suitable for students in Grades 9-12.

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