The second term of an arithmetic sequence is 24 and the 5th term is 3 Find the first term

Given the second term \( a_2 = 24 \) and the fifth term \( a_5 = 3 \) of an arithmetic sequence, we need to find the first term \( a_1 \).

Let's denote the first term by \( a \) and the common difference by \( d \).

The formula for the \( n \)-th term of an arithmetic sequence is:

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

Using the given information:

1. For the second term (\( a_2 \)):
   \[ a + 1d = 24 \]
   \[ a + d = 24 \quad \text{(Equation 1)} \]

2. For the fifth term (\( a_5 \)):
   \[ a + 4d = 3 \quad \text{(Equation 2)} \]

We now have two equations:
1. \( a + d = 24 \)
2. \( a + 4d = 3 \)

To find \( a \) and \( d \), we can solve this system of equations.

First, subtract Equation 1 from Equation 2:

\[ (a + 4d) - (a + d) = 3 - 24 \]
\[ a + 4d - a - d = 3 - 24 \]
\[ 3d = -21 \]
\[ d = -7 \]

Now, substitute \( d = -7 \) back into Equation 1:

\[ a + (-7) = 24 \]
\[ a - 7 = 24 \]
\[ a = 31 \]

Therefore, the first term \( a_1 \) is \( 31 \).

### Summary:
The first term of the arithmetic sequence is \( \boxed{31} \).

Would you like more details or have any questions? Here are some related questions you might consider:

1. How do you derive the general formula for the \( n \)-th term of an arithmetic sequence?
2. How can you find the common difference if more terms of the sequence are given?
3. What is the sum of the first \( n \) terms of an arithmetic sequence?
4. How does changing the common difference affect the sequence?
5. Can an arithmetic sequence have a common difference of zero? What would it look like?
6. How can you determine if a sequence is arithmetic just by looking at a few terms?
7. How would you solve for the 10th term in the given sequence?
8. How can you graphically represent an arithmetic sequence?

**Tip:** Always write down the equations you derive from the problem, as this makes it easier to follow the logical steps to the solution.

Learn how to find the first term of an arithmetic sequence when the second term is 24 and the fifth term is 3. This problem involves using the formula for the n-th term of an arithmetic sequence and solving a system of equations. Suitable for students in Grades 9-12.

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