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) given that the 7th term is 6 and the 21st term is -22, we can follow these steps:

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

For the 7th term:
\[ a_7 = a + 6d = 6 \quad \text{(1)} \]

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

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

We can subtract equation (1) from equation (2):
\[ (a + 20d) - (a + 6d) = -22 - 6 \]
This simplifies to:
\[ 14d = -28 \]
Now, divide both sides by 14:
\[ d = -2 \]

### Step 3: Substitute \( d \) back to find \( a \)
Now that we have \( d \), we can substitute \( d = -2 \) back into equation (1):
\[ a + 6(-2) = 6 \]
This simplifies to:
\[ a - 12 = 6 \]
Adding 12 to both sides gives:
\[ a = 18 \]

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

### Final Answer
The 26th term of the arithmetic progression is:
\[ \boxed{-32} \]

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