Let's find the first term of the arithmetic progression given the following information:

• Common difference ($d$) = $-5$
• Sum of the first 10 terms ($S_{10}$) = 45

Step 1: Use the formula for the sum of the first $n$ terms

The sum of the first $n$ terms of an arithmetic progression is given by: $S_n = \frac{n}{2} \times \left(2a_1 + (n-1)d\right)$ For $n = 10$, $d = -5$, and $S_{10} = 45$: $S_{10} = \frac{10}{2} \times \left(2a_1 + (10-1)(-5)\right)$ $45 = 5 \times \left(2a_1 + 9(-5)\right)$

Step 2: Simplify the equation

$45 = 5 \times \left(2a_1 - 45\right)$ $45 = 10a_1 - 225$

Step 3: Solve for $a_1$

Add 225 to both sides: $270 = 10a_1$ Divide by 10: $a_1 = 27$

Final Answer:

The first term of the arithmetic progression is 27.

Would you like more details or have any questions?

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Here are some related questions you might be interested in:

1. How does the negative common difference affect the progression?
2. Can you calculate the sum of the first 20 terms with the same first term and common difference?
3. How do you find the first term if the sum and the last term are given?
4. What is the relationship between the sum, first term, and common difference?
5. How can you determine the number of terms if the sum and common difference are known?
6. What happens to the sum if the common difference is positive instead of negative?
7. How would you find the common difference if only the sum and the first term are given?
8. Can you find the $n$th term of this progression?

Tip: When dealing with negative common differences, the sequence decreases, which can sometimes result in negative sums if the sequence is long enough.