Find the A.P. whose 6th term =5 and 10th term=9

Answer

To find the arithmetic progression (A.P.) whose 6th term is 5 and 10th term is 9, we can follow these steps:

### Step 1: Understand the terms of an A.P.
In an arithmetic progression, the nth term (denoted as \( a_n \)) can be expressed as:
\[ a_n = a + (n-1)d \]
where \( a \) is the first term and \( d \) is the common difference.

### Step 2: Write equations for the given terms
We are given:
- The 6th term \( a_6 = 5 \)
- The 10th term \( a_{10} = 9 \)

Using the formula for the nth term, we can write:
1. For the 6th term:
\[ a + 5d = 5 \quad \text{(Equation 1)} \]
2. For the 10th term:
\[ a + 9d = 9 \quad \text{(Equation 2)} \]

### Step 3: Set up the equations
Now we have two equations:
1. \( a + 5d = 5 \)
2. \( a + 9d = 9 \)

### Step 4: Subtract Equation 1 from Equation 2
To eliminate \( a \), we subtract Equation 1 from Equation 2:
\[ (a + 9d) - (a + 5d) = 9 - 5 \]
This simplifies to:
\[ 4d = 4 \]

### Step 5: Solve for \( d \)
Now, divide both sides by 4:
\[ d = 1 \]

### Step 6: Substitute \( d \) back into Equation 1
Now that we have \( d \), we can substitute it back into Equation 1 to find \( a \):
\[ a + 5(1) = 5 \]
This simplifies to:
\[ a + 5 = 5 \]
Subtract 5 from both sides:
\[ a = 0 \]

### Step 7: Write the A.P.
Now that we have both \( a \) and \( d \):
- First term \( a = 0 \)
- Common difference \( d = 1 \)

The A.P. can be written as:
\[ 0, 1, 2, 3, 4, \ldots \]

### Final Answer
The arithmetic progression is:
\[ 0, 1, 2, 3, 4, \ldots \]

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