What is the 18th member of an aritmetic progression with it's second member equaling 1 and the sum of the first 12 members equaling -42

To solve for the 18th term of an arithmetic progression, we need to follow these steps:

### Step 1: General formula for an arithmetic progression
The \(n\)-th term of an arithmetic progression 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 term number.

### Step 2: Use the second term condition
We are told that the second term is 1, so:

\[
a_2 = a_1 + (2 - 1) \cdot d = a_1 + d
\]

Given \(a_2 = 1\), we have:

\[
a_1 + d = 1 \quad \text{(Equation 1)}
\]

### Step 3: Use the sum of the first 12 terms condition
The sum of the first \(n\) terms of an arithmetic progression is given by the formula:

\[
S_n = \frac{n}{2} \cdot (2a_1 + (n-1) \cdot d)
\]

For the sum of the first 12 terms (\(S_{12} = -42\)):

\[
S_{12} = \frac{12}{2} \cdot (2a_1 + (12 - 1) \cdot d) = -42
\]

This simplifies to:

\[
6 \cdot (2a_1 + 11d) = -42
\]

\[
2a_1 + 11d = -7 \quad \text{(Equation 2)}
\]

### Step 4: Solve the system of equations
We now have two equations:
1. \(a_1 + d = 1\)
2. \(2a_1 + 11d = -7\)

#### Solve Equation 1 for \(a_1\):
From Equation 1, we get:

\[
a_1 = 1 - d
\]

#### Substitute this into Equation 2:
\[
2(1 - d) + 11d = -7
\]

\[
2 - 2d + 11d = -7
\]

\[
2 + 9d = -7
\]

\[
9d = -9
\]

\[
d = -1
\]

### Step 5: Find \(a_1\)
Using \(d = -1\) in Equation 1:

\[
a_1 + (-1) = 1
\]

\[
a_1 = 2
\]

### Step 6: Find the 18th term
Now that we have \(a_1 = 2\) and \(d = -1\), we can find the 18th term using the formula for the \(n\)-th term:

\[
a_{18} = a_1 + (18 - 1) \cdot d = 2 + 17 \cdot (-1)
\]

\[
a_{18} = 2 - 17 = -15
\]

### Final Answer:
The 18th term of the arithmetic progression is \(-15\).

Would you like further details or have any questions?

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Here are 5 related questions to explore further:
1. How do you find the common difference in an arithmetic progression?
2. What is the formula for the sum of an arithmetic series?
3. How would the answer change if the sum of the first 12 terms were different?
4. Can you find the general form of the arithmetic sequence in this case?
5. How does the position of a term affect its value in an arithmetic progression?

**Tip:** Always verify your arithmetic series formulas with a quick check to avoid errors.

What is the 18th member of an arithmetic progression with its second member equaling 1 and the sum of the first 12 members equaling -42?

This problem helps students learn how to find the 18th term of an arithmetic progression given specific conditions: the second term equals 1, and the sum of the first 12 terms is -42. It involves key algebraic formulas and arithmetic sequence concepts. Suitable for Grades 9-12.

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