The sum of n terms of a series is n(n+1) . Prove that it is an A.P. also find its 10th term.

Answer

To prove that the given series is an Arithmetic Progression (A.P.) and to find its 10th term, we can follow these steps:

### Step 1: Define the sum of n terms
Let the sum of the first n terms of the series be denoted as \( S_n \). According to the problem, we have:
\[ S_n = n(n + 1) \]

### Step 2: Use the formula for the sum of n terms of an A.P.
The formula for the sum of the first n terms of an A.P. is given by:
\[ S_n = \frac{n}{2} \left(2A + (n - 1)d\right) \]
where \( A \) is the first term and \( d \) is the common difference.

### Step 3: Set the two expressions for \( S_n \) equal
Equating the two expressions for \( S_n \):
\[ n(n + 1) = \frac{n}{2} \left(2A + (n - 1)d\right) \]

### Step 4: Simplify the equation
Multiply both sides by 2 to eliminate the fraction:
\[ 2n(n + 1) = n(2A + (n - 1)d) \]
Assuming \( n \neq 0 \), we can divide both sides by \( n \):
\[ 2(n + 1) = 2A + (n - 1)d \]

### Step 5: Rearrange the equation
Rearranging gives:
\[ 2A + (n - 1)d = 2n + 2 \]
This can be rewritten as:
\[ (n - 1)d = 2n + 2 - 2A \]

### Step 6: Analyze the coefficients
To show that the sequence is an A.P., we need to show that \( d \) is constant. We can analyze the equation:
1. For \( n = 1 \):
\[ 0 \cdot d = 2 \cdot 1 + 2 - 2A \implies 0 = 4 - 2A \implies A = 2 \]
2. For \( n = 2 \):
\[ 1 \cdot d = 2 \cdot 2 + 2 - 2 \cdot 2 \implies d = 4 + 2 - 4 \implies d = 2 \]

### Step 7: Conclusion about A.P.
Since we found that \( A = 2 \) and \( d = 2 \), the series is indeed an A.P. with first term \( A = 2 \) and common difference \( d = 2 \).

### Step 8: Find the 10th term
The nth term of an A.P. can be calculated using the formula:
\[ A_n = A + (n - 1)d \]
For the 10th term \( A_{10} \):
\[ A_{10} = 2 + (10 - 1) \cdot 2 = 2 + 9 \cdot 2 = 2 + 18 = 20 \]

### Final Answer
Thus, the 10th term of the series is:
\[ \boxed{20} \]

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