Determine 2nd term and 5'th term of an A.P. whose 6th term is 12 and 8th term is 22

Answer

To determine the 2nd term and 5th term of an arithmetic progression (A.P.) where the 6th term is 12 and the 8th term is 22, we can follow these steps:

### Step 1: Use the formula for the nth term of an A.P.
The nth term of an A.P. can be expressed as:
\[ T_n = a + (n - 1) \cdot d \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Set up equations for the 6th and 8th terms
From the information given:
- For the 6th term:
\[ T_6 = a + (6 - 1) \cdot d = 12 \]
This simplifies to:
\[ a + 5d = 12 \quad \text{(Equation 1)} \]

- For the 8th term:
\[ T_8 = a + (8 - 1) \cdot d = 22 \]
This simplifies to:
\[ a + 7d = 22 \quad \text{(Equation 2)} \]

### Step 3: Solve the equations simultaneously
Now we have two equations:
1. \( a + 5d = 12 \)
2. \( a + 7d = 22 \)

To eliminate \( a \), we can subtract Equation 1 from Equation 2:
\[ (a + 7d) - (a + 5d) = 22 - 12 \]
This simplifies to:
\[ 2d = 10 \]
So, we find:
\[ d = 5 \]

### Step 4: Substitute \( d \) back to find \( a \)
Now substitute \( d = 5 \) back into Equation 1:
\[ a + 5 \cdot 5 = 12 \]
This simplifies to:
\[ a + 25 = 12 \]
Thus,
\[ a = 12 - 25 = -13 \]

### Step 5: Find the 2nd term \( T_2 \)
Using the formula for the 2nd term:
\[ T_2 = a + (2 - 1) \cdot d = a + d \]
Substituting the values of \( a \) and \( d \):
\[ T_2 = -13 + 5 = -8 \]

### Step 6: Find the 5th term \( T_5 \)
Using the formula for the 5th term:
\[ T_5 = a + (5 - 1) \cdot d = a + 4d \]
Substituting the values of \( a \) and \( d \):
\[ T_5 = -13 + 4 \cdot 5 = -13 + 20 = 7 \]

### Final Answer
Thus, the 2nd term \( T_2 \) is \(-8\) and the 5th term \( T_5 \) is \(7\).