For an arithmetic sequence, the terms a8=21 and a10=25 are known. Find the first term a1.

To find the first term \( a_1 \) of the arithmetic sequence, we need to use the information given about the terms \( a_8 \) and \( a_{10} \).

Given:
\[ a_8 = 21 \]
\[ a_{10} = 25 \]

The general formula for the \( n \)-th term of an arithmetic sequence is:
\[ a_n = a_1 + (n-1)d \]

where \( a_1 \) is the first term and \( d \) is the common difference.

First, let's write the equations for \( a_8 \) and \( a_{10} \) using the general formula:

For \( a_8 \):
\[ a_8 = a_1 + 7d = 21 \]

For \( a_{10} \):
\[ a_{10} = a_1 + 9d = 25 \]

We now have a system of linear equations:
1. \( a_1 + 7d = 21 \)
2. \( a_1 + 9d = 25 \)

To solve for \( a_1 \) and \( d \), we can subtract the first equation from the second:

\[ (a_1 + 9d) - (a_1 + 7d) = 25 - 21 \]
\[ 2d = 4 \]
\[ d = 2 \]

Now that we have the common difference \( d \), we can substitute it back into one of the original equations to find \( a_1 \):

Using \( a_1 + 7d = 21 \):
\[ a_1 + 7 \cdot 2 = 21 \]
\[ a_1 + 14 = 21 \]
\[ a_1 = 21 - 14 \]
\[ a_1 = 7 \]

Therefore, the first term \( a_1 \) of the arithmetic sequence is \( 7 \).

Would you like any further details or have any questions about this solution?

Here are 8 related questions to further expand on the topic:

1. How would you find the common difference if the terms \( a_8 \) and \( a_{10} \) were different?
2. Can you derive the \( n \)-th term formula for an arithmetic sequence?
3. How do you find the sum of the first \( n \) terms of an arithmetic sequence?
4. What is the relationship between the common difference and the sequence's growth rate?
5. How can you determine the number of terms in an arithmetic sequence if the first term, last term, and common difference are known?
6. How do you solve for a specific term in an arithmetic sequence given the first term and the common difference?
7. How would the solution change if the sequence were geometric instead of arithmetic?
8. What are some real-world applications of arithmetic sequences?

**Tip:** When solving systems of equations, always check your solution by substituting the values back into the original equations to ensure they hold true.

Learn how to find the first term of an arithmetic sequence using given terms a8=21 and a10=25. Step-by-step solution involving arithmetic sequences and linear equations. Suitable for high school students and advanced learners.

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