The nth term of a progression is 2n+3. Show that it is an A.P. Also find is 10th term.

Explanation

To solve the problem step by step, we need to show that the given sequence defined by the nth term \( T_n = 2n + 3 \) is an arithmetic progression (A.P.) and then find the 10th term of the sequence.

### Step 1: Define the nth term
The nth term of the progression is given as:
\[ T_n = 2n + 3 \]

### Step 2: Find the (n-1)th term
To show that the sequence is an A.P., we need to find the (n-1)th term:
\[ T_{n-1} = 2(n-1) + 3 \]
Calculating this gives:
\[ T_{n-1} = 2n - 2 + 3 = 2n + 1 \]

### Step 3: Calculate the difference between consecutive terms
Now, we calculate the difference between the nth term and the (n-1)th term:
\[ T_n - T_{n-1} = (2n + 3) - (2n + 1) \]
Simplifying this expression:
\[ T_n - T_{n-1} = 2n + 3 - 2n - 1 = 2 \]

### Step 4: Conclusion about the A.P.
Since the difference \( T_n - T_{n-1} = 2 \) is constant (does not depend on n), we can conclude that the sequence is indeed an arithmetic progression (A.P.).

### Step 5: Find the 10th term
To find the 10th term, we substitute \( n = 10 \) into the nth term formula:
\[ T_{10} = 2(10) + 3 \]
Calculating this gives:
\[ T_{10} = 20 + 3 = 23 \]

### Final Answer
The 10th term of the progression is:
\[ \boxed{23} \]