find the sum `3+5+7+`…………. upto n terms is

Options

A. `n^(2)`
B. `n(n-2)`
C. `n(n+2)`
D. `(n+1)^(2)`

Answer

To find the sum of the series \(3 + 5 + 7 + \ldots\) up to \(n\) terms, we can follow these steps:

### Step 1: Identify the first term and common difference
The given series is an arithmetic progression (AP).
- The first term \(A = 3\)
- The common difference \(D = 5 - 3 = 2\)

### Step 2: Write the formula for the sum of the first \(n\) terms of an AP
The formula for the sum \(S_n\) of the first \(n\) terms of an arithmetic progression is given by:
\[ S_n = \frac{n}{2} \times (2A + (n - 1)D) \]

### Step 3: Substitute the values of \(A\) and \(D\) into the formula
Now, substituting \(A = 3\) and \(D = 2\) into the formula:
\[ S_n = \frac{n}{2} \times (2 \times 3 + (n - 1) \times 2) \]

### Step 4: Simplify the expression
Now, simplify the expression:
\[ S_n = \frac{n}{2} \times (6 + 2(n - 1)) \]
\[ = \frac{n}{2} \times (6 + 2n - 2) \]
\[ = \frac{n}{2} \times (2n + 4) \]
\[ = \frac{n}{2} \times 2(n + 2) \]
\[ = n(n + 2) \]

### Final Answer
Thus, the sum of the series \(3 + 5 + 7 + \ldots\) up to \(n\) terms is:
\[ S_n = n(n + 2) \]

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