The problem you uploaded is related to an arithmetic sequence with the following information:

• Common difference $d = -1$
• The sum of the first 10 terms $S_{10} = 45$
• You are asked to find the first term $a_1$ and the general term formula $a_n$.

Step 1: Use the sum formula for an arithmetic sequence

The sum of the first $n$ terms of an arithmetic sequence is given by the formula: $S_n = \frac{n}{2} \times (2a_1 + (n - 1)d)$ For $S_{10} = 45$, $n = 10$, and $d = -1$, we substitute these values into the sum formula: $45 = \frac{10}{2} \times (2a_1 + (10 - 1)(-1))$ Simplifying: $45 = 5 \times (2a_1 + 9(-1))$ $45 = 5 \times (2a_1 - 9)$ $45 = 10a_1 - 45$

Step 2: Solve for $a_1$

Add 45 to both sides: $90 = 10a_1$ Divide by 10: $a_1 = 9$

Step 3: Find the general term formula

The general term of an arithmetic sequence is given by: $a_n = a_1 + (n - 1)d$ Substituting $a_1 = 9$ and $d = -1$: $a_n = 9 + (n - 1)(-1)$ Simplifying: $a_n = 9 - n + 1$ $a_n = 10 - n$

Final Answers:

• The first term $a_1 = 9$
• The general term formula is $a_n = 10 - n$

Would you like more details or have any questions?

Here are 5 related questions:

1. How does the common difference affect the terms of an arithmetic sequence?
2. What happens if the common difference is positive instead of negative?
3. How can you derive the sum formula for an arithmetic sequence?
4. How would the general term formula change if the sum of the first 20 terms were given instead of 10?
5. What is the 15th term of the sequence using the formula $a_n = 10 - n$?

Tip: In arithmetic sequences, the general term formula directly depends on the common difference and the first term. Knowing these two can help you find any term in the sequence.