Find the 20th, 25th and nth term of the A.P. Given by
21, 16,11,6,1,-4,-9……………………..

Answer

To solve the problem of finding the 20th, 25th, and nth term of the arithmetic progression (A.P.) given by the sequence 21, 16, 11, 6, 1, -4, -9, we can follow these steps:

### Step 1: Identify the first term (A) and the common difference (D)
The first term \( A \) of the A.P. is the first number in the sequence:
\[ A = 21 \]

Next, we calculate the common difference \( D \) by subtracting the first term from the second term:
\[ D = 16 - 21 = -5 \]

### Step 2: Write the formula for the nth term of the A.P.
The formula for the nth term \( T_n \) of an A.P. is given by:
\[ T_n = A + (n - 1) \cdot D \]

Substituting the values of \( A \) and \( D \):
\[ T_n = 21 + (n - 1)(-5) \]

### Step 3: Simplify the formula for the nth term
Now, we simplify the expression:
\[ T_n = 21 - 5(n - 1) \]
\[ T_n = 21 - 5n + 5 \]
\[ T_n = 26 - 5n \]

### Step 4: Find the 20th term (T20)
Now, we can find the 20th term by substituting \( n = 20 \) into the formula:
\[ T_{20} = 26 - 5 \cdot 20 \]
\[ T_{20} = 26 - 100 \]
\[ T_{20} = -74 \]

### Step 5: Find the 25th term (T25)
Next, we find the 25th term by substituting \( n = 25 \):
\[ T_{25} = 26 - 5 \cdot 25 \]
\[ T_{25} = 26 - 125 \]
\[ T_{25} = -99 \]

### Step 6: Write the final formula for the nth term
The nth term of the A.P. is:
\[ T_n = 26 - 5n \]

### Summary of Results
- The 20th term \( T_{20} \) is \( -74 \).
- The 25th term \( T_{25} \) is \( -99 \).
- The nth term \( T_n \) is \( 26 - 5n \).