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 $(-1)^{k+1}$.

(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:

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

(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:

$\sum_{n=1}^{27} a_{3n}$

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

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.