What is answer for a sub 5= 31, a sub 10= 1, what is a sub 1?

To find \( a_1 \) given \( a_5 = 31 \) and \( a_{10} = 1 \), we need to assume a specific type of sequence. Let's consider the possibility of an arithmetic sequence, where the difference between consecutive terms is constant.

In an arithmetic sequence, the general formula for the \( n \)-th term is:

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

Where:
- \( a_n \) is the \( n \)-th term,
- \( a_1 \) is the first term,
- \( d \) is the common difference.

### Step 1: Use the formula for \( a_5 = 31 \)

\[
a_5 = a_1 + (5 - 1)d = a_1 + 4d = 31
\]

### Step 2: Use the formula for \( a_{10} = 1 \)

\[
a_{10} = a_1 + (10 - 1)d = a_1 + 9d = 1
\]

### Step 3: Solve the system of equations

We have two equations:
1. \( a_1 + 4d = 31 \)
2. \( a_1 + 9d = 1 \)

Subtract the first equation from the second:

\[
(a_1 + 9d) - (a_1 + 4d) = 1 - 31
\]
\[
5d = -30
\]
\[
d = -6
\]

### Step 4: Substitute \( d = -6 \) back into the first equation

\[
a_1 + 4(-6) = 31
\]
\[
a_1 - 24 = 31
\]
\[
a_1 = 55
\]

### Answer:
The first term \( a_1 \) of the sequence is \( 55 \).

Would you like more details or have any questions?

Here are 5 questions to consider:

1. How would the answer change if the sequence were geometric instead of arithmetic?
2. What is the common difference in an arithmetic sequence?
3. Can you find the value of the 15th term \( a_{15} \) in this sequence?
4. What are the first five terms of this sequence?
5. How can you verify that your sequence is correct?

**Tip:** When dealing with sequences, setting up and solving a system of equations is a powerful method to determine unknown terms, especially in arithmetic or geometric sequences.

Learn how to find the first term of an arithmetic sequence given the 5th term is 31 and the 10th term is 1. Step-by-step solution using arithmetic sequence formula and system of linear equations. Suitable for high school students.

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