Find the sum of 1+3 + 5 +7 + … 21

Answer

To find the sum of the series 1 + 3 + 5 + 7 + ... + 21, we can follow these steps:

### Step 1: Identify the series
The series consists of odd numbers starting from 1 and ending at 21. This is an arithmetic series where:
- The first term (A) is 1.
- The common difference (D) is 2 (since each term increases by 2).

### Step 2: Determine the number of terms (n)
To find the number of terms in the series, we can use the formula for the nth term of an arithmetic sequence:
\[ T_n = A + (n - 1) \cdot D \]
We need to find n when \( T_n = 21 \):
\[ 21 = 1 + (n - 1) \cdot 2 \]

### Step 3: Solve for n
Rearranging the equation:
\[ 21 - 1 = (n - 1) \cdot 2 \]
\[ 20 = (n - 1) \cdot 2 \]
Dividing both sides by 2:
\[ 10 = n - 1 \]
Adding 1 to both sides:
\[ n = 11 \]
So, there are 11 terms in the series.

### Step 4: Use the formula for the sum of an arithmetic series
The formula for the sum (S) of the first n terms of an arithmetic series is:
\[ S_n = \frac{n}{2} \cdot (2A + (n - 1) \cdot D) \]

### Step 5: Substitute the values into the formula
Now substituting the values we found:
- \( n = 11 \)
- \( A = 1 \)
- \( D = 2 \)

\[ S_{11} = \frac{11}{2} \cdot (2 \cdot 1 + (11 - 1) \cdot 2) \]
\[ S_{11} = \frac{11}{2} \cdot (2 + 20) \]
\[ S_{11} = \frac{11}{2} \cdot 22 \]

### Step 6: Calculate the sum
\[ S_{11} = \frac{11 \cdot 22}{2} \]
\[ S_{11} = \frac{242}{2} \]
\[ S_{11} = 121 \]

Thus, the sum of the series 1 + 3 + 5 + 7 + ... + 21 is **121**.

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