The nth term of a sequence is 8 -5n. Show that the sequence is an A.P.

Answer

To show that the sequence defined by the nth term \( T(n) = 8 - 5n \) is an arithmetic progression (A.P.), we will follow these steps:

### Step 1: Write the nth term
The nth term of the sequence is given as:
\[ T(n) = 8 - 5n \]

### Step 2: Find the (n+1)th term
To find the next term in the sequence, we will replace \( n \) with \( n + 1 \):
\[ T(n+1) = 8 - 5(n + 1) = 8 - 5n - 5 = 3 - 5n \]

### Step 3: Calculate the difference between consecutive terms
Now, we will find the difference between the (n+1)th term and the nth term:
\[ T(n+1) - T(n) = (3 - 5n) - (8 - 5n) \]

### Step 4: Simplify the difference
Now, we simplify the expression:
\[ T(n+1) - T(n) = 3 - 5n - 8 + 5n \]
The \( -5n \) and \( +5n \) cancel each other out:
\[ T(n+1) - T(n) = 3 - 8 = -5 \]

### Step 5: Conclusion
The difference \( T(n+1) - T(n) = -5 \) is a constant value. Since the difference between consecutive terms is constant, we can conclude that the sequence is an arithmetic progression (A.P.).

Thus, we have shown that the sequence defined by \( T(n) = 8 - 5n \) is an A.P.