Find the nth term and 20th term of the sequance : 9,5,1,1-3………

Answer

To find the nth term and the 20th term of the sequence 9, 5, 1, -3, we can follow these steps:

### Step 1: Identify the first term (a) and the common difference (d)
The first term \( a \) of the sequence is:
\[ a = 9 \]

To find the common difference \( d \), we subtract the first term from the second term:
\[ d = a_2 - a_1 = 5 - 9 = -4 \]

### Step 2: Write the formula for the nth term
The formula for the nth term of an arithmetic progression (AP) is given by:
\[ a_n = a + (n - 1) \cdot d \]

### Step 3: Substitute the values of a and d into the formula
Substituting \( a = 9 \) and \( d = -4 \) into the formula:
\[ a_n = 9 + (n - 1) \cdot (-4) \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ a_n = 9 - 4(n - 1) \]
\[ a_n = 9 - 4n + 4 \]
\[ a_n = 13 - 4n \]

Thus, the nth term of the sequence is:
\[ a_n = 13 - 4n \]

### Step 5: Find the 20th term
To find the 20th term, substitute \( n = 20 \) into the nth term formula:
\[ a_{20} = 13 - 4 \cdot 20 \]
\[ a_{20} = 13 - 80 \]
\[ a_{20} = -67 \]

### Final Answer
The nth term of the sequence is \( 13 - 4n \) and the 20th term is \( -67 \).

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