We are given that $u_6 = -5$ and $u_9 = -20$ in an arithmetic sequence, and we need to find $S_{20}$, the sum of the first 20 terms.

Step 1: General formula for an arithmetic sequence

The general term $u_n$ of an arithmetic sequence is given by:

$u_n = u_1 + (n - 1) d$

Where:

• $u_n$ is the $n$-th term,
• $u_1$ is the first term,
• $d$ is the common difference.

Step 2: Use the information provided

We know:

$u_6 = u_1 + 5d = -5$ $u_9 = u_1 + 8d = -20$

Now, subtract the first equation from the second to eliminate $u_1$:

$(u_1 + 8d) - (u_1 + 5d) = -20 - (-5)$ $3d = -15$ $d = -5$

Step 3: Find $u_1$

Substitute $d = -5$ into one of the original equations, say $u_6 = u_1 + 5d = -5$:

$u_1 + 5(-5) = -5$ $u_1 - 25 = -5$ $u_1 = 20$

Step 4: Find $S_{20}$

The sum of the first $n$ terms in an arithmetic sequence is given by:

$S_n = \frac{n}{2} \times (u_1 + u_n)$

We already know $u_1 = 20$. To find $u_{20}$:

$u_{20} = u_1 + 19d = 20 + 19(-5) = 20 - 95 = -75$

Now, calculate $S_{20}$:

$S_{20} = \frac{20}{2} \times (20 + (-75)) = 10 \times (-55) = -550$

Final Answer:

$S_{20} = -550$

Would you like more details on any part of this solution?

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Here are five related questions to expand your understanding:

1. How do you find the common difference in an arithmetic sequence when given two terms?
2. How can you derive the sum formula for an arithmetic sequence?
3. What happens to the sum if the common difference is positive rather than negative?
4. How do you find the sum of an arithmetic sequence if only partial terms are given?
5. How is the sum formula affected when the sequence starts from a different term than $u_1$?

Tip: Always double-check your calculations when using the sum formula, especially when dealing with negative numbers.