For a given arithmetic sequence, the common difference, 

*d*

, is equal to 

−8

, and the 

77th

 term, 

*a*77

, is equal to 

−626

.


To solve this problem, we need to find the first term (\(a_1\)) of the arithmetic sequence. The formula for the \(n\)-th term of an arithmetic sequence is given by:

\[
a_n = a_1 + (n - 1) \cdot d
\]

where:
- \(a_n\) is the \(n\)-th term,
- \(a_1\) is the first term,
- \(d\) is the common difference,
- \(n\) is the position of the term in the sequence.

We are given:
- The common difference \(d = -8\),
- The 77th term \(a_{77} = -626\),
- \(n = 77\).

Substituting the given values into the formula:

\[
a_{77} = a_1 + (77 - 1) \cdot (-8)
\]
\[
-626 = a_1 + 76 \cdot (-8)
\]
\[
-626 = a_1 - 608
\]
\[
a_1 = -626 + 608
\]
\[
a_1 = -18
\]

Thus, the first term of the arithmetic sequence is \(-18\).

### Summary:
- Common difference, \(d = -8\)
- First term, \(a_1 = -18\)
- 77th term, \(a_{77} = -626\)

Would you like further details or have any questions?

### Related questions:
1. What is the 100th term of this sequence?
2. How can you find the sum of the first 77 terms of this sequence?
3. What would be the 50th term of the sequence?
4. How do you find the general formula for the \(n\)-th term of an arithmetic sequence?
5. If another arithmetic sequence had the same common difference but a different first term, how would it affect the terms?

### Tip:
To quickly calculate any term in an arithmetic sequence, use the formula \(a_n = a_1 + (n - 1) \cdot d\), where the first term and common difference are key!

For a given arithmetic sequence, the common difference, d, is equal to -8, and the 77th term, a_77, is equal to -626.

Learn how to determine the first term of an arithmetic sequence where the common difference is -8 and the 77th term is -626. This problem covers key concepts like arithmetic sequences and the formula for the nth term, suitable for students in Grades 9-10.

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