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

Question

Accepted Answer

To determine the 26th term of the arithmetic progression (A.P.), we need to first establish the common difference and the first term of the sequence using the information provided about the 7th and 21st terms.

Given:
- The 7th term of the A.P. is 6.
- The 21st term of the A.P. is -22.

The formula for the nth term of an A.P. is given by:
$$ a_n = a + (n-1) \cdot d $$
where $ a $ is the first term and $ d $ is the common difference.

1. **Finding the common difference ($d$):**

Using the formula for the 7th term:
   $$ a + 6d = 6 $$  (Equation 1)

Using the formula for the 21st term:
   $$ a + 20d = -22 $$  (Equation 2)

Subtract Equation 1 from Equation 2 to eliminate $a$:
   $$
   (a + 20d) - (a + 6d) = -22 - 6 \
   14d = -28 \
   d = -2
   $$

2. **Finding the first term ($a$):**

Substitute $d = -2$ back into Equation 1:
   $$
   a + 6(-2) = 6 \
   a - 12 = 6 \
   a = 18
   $$

3. **Finding the 26th term:**

Now that we have $a = 18$ and $d = -2$, we can find the 26th term:
   $$
   a_{26} = a + 25d \
   a_{26} = 18 + 25(-2) \
   a_{26} = 18 - 50 \
   a_{26} = -32
   $$

Thus, the 26th term of the A.P. is $-32$, which corresponds to Option 1.

**Explanation for Incorrect Options:**

- **Option 2 (-34):** This value does not satisfy the calculation for the 26th term using the derived values of $a$ and $d$.
- **Option 3 (-30):** This value is incorrect as it does not match the calculated 26th term.
- **Option 4 (-36):** This value is also incorrect based on the arithmetic progression calculation.

Therefore, the correct answer is indeed Option 1, $-32$.