In an arithmetic sequence, we are given:

• $U_6 = -5$
• $U_9 = -20$

We need to find $S_{20}$, the sum of the first 20 terms of the sequence.

Step 1: Use the formula for the general term of an arithmetic sequence

The formula for the $n$-th term of an arithmetic sequence is:

$U_n = U_1 + (n-1) \cdot d$

Where:

• $U_n$ is the $n$-th term,
• $U_1$ is the first term,
• $d$ is the common difference.

For $U_6$ and $U_9$:

$U_6 = U_1 + 5d = -5 \quad \text{(Equation 1)}$ $U_9 = U_1 + 8d = -20 \quad \text{(Equation 2)}$

Step 2: Solve the system of equations

Subtract Equation 1 from Equation 2 to eliminate $U_1$:

$(U_1 + 8d) - (U_1 + 5d) = -20 - (-5)$ $3d = -15$ $d = -5$

Step 3: Find $U_1$

Substitute $d = -5$ back into Equation 1:

$U_1 + 5(-5) = -5$ $U_1 - 25 = -5$ $U_1 = 20$

So the first term $U_1 = 20$ and the common difference $d = -5$.

Step 4: Find $S_{20}$

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

$S_n = \frac{n}{2} \cdot (U_1 + U_n)$

To find $S_{20}$, we need $U_{20}$, which is:

$U_{20} = U_1 + (20-1) \cdot d = 20 + 19(-5) = 20 - 95 = -75$

Now, substitute into the sum formula:

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

Final Answer:

$S_{20} = -550$

Would you like further details or have any questions?

Here are 5 related questions:

1. How do you find the general term of an arithmetic sequence?
2. How is the sum of an arithmetic sequence derived?
3. What happens to the sum if the common difference is positive instead of negative?
4. Can you solve for the sum if $S_{15}$ is also given?
5. How does changing the first term affect the sum of an arithmetic sequence?

Tip: In arithmetic sequences, knowing any two terms allows you to find the common difference, which is key to solving other parts of the sequence.