The 7th and 21st terms of an arithmetic progression are 6 and -22 respectively, 26th term of A.P. is equal to:

Options

A. -32
B. -34
C. -30
D. -36

Answer

To find the 26th term of the arithmetic progression (AP), we start by using the information given about the 7th and 21st terms.

### Step 1: Set up the equations for the 7th and 21st terms
The formula for the nth term of an arithmetic progression is given by:
\[ A_n = a + (n-1)d \]
where:
- \( A_n \) is the nth term,
- \( a \) is the first term,
- \( d \) is the common difference,
- \( n \) is the term number.

For the 7th term:
\[ A_7 = a + (7-1)d = a + 6d = 6 \quad \text{(1)} \]

For the 21st term:
\[ A_{21} = a + (21-1)d = a + 20d = -22 \quad \text{(2)} \]

### Step 2: Solve the equations simultaneously
Now we have two equations:
1. \( a + 6d = 6 \) (Equation 1)
2. \( a + 20d = -22 \) (Equation 2)

We can subtract Equation 1 from Equation 2 to eliminate \( a \):
\[ (a + 20d) - (a + 6d) = -22 - 6 \]
\[ 14d = -28 \]
\[ d = -2 \]

### Step 3: Substitute \( d \) back to find \( a \)
Now that we have \( d \), we can substitute it back into Equation 1 to find \( a \):
\[ a + 6(-2) = 6 \]
\[ a - 12 = 6 \]
\[ a = 18 \]

### Step 4: Find the 26th term
Now we can find the 26th term using the formula:
\[ A_{26} = a + (26-1)d = a + 25d \]
Substituting the values of \( a \) and \( d \):
\[ A_{26} = 18 + 25(-2) \]
\[ A_{26} = 18 - 50 \]
\[ A_{26} = -32 \]

### Final Answer
The 26th term of the arithmetic progression is \(-32\).

---