Find the sum of odd numbers from 1 to 31.

Options

A. 341
B. 248
C. 321
D. 256

Answer

To find the sum of odd numbers from 1 to 31, we can follow these steps:

### Step 1: Identify the odd numbers
The odd numbers from 1 to 31 are:
1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 31.

### Step 2: Recognize the pattern
These numbers form an arithmetic progression (AP) where:
- The first term (A) is 1.
- The common difference (D) is 2.
- The last term (L) is 31.

### Step 3: Find the number of terms (n)
We can use the formula for the nth term of an arithmetic progression:
\[ L = A + (n - 1) \cdot D \]
Substituting the known values:
\[ 31 = 1 + (n - 1) \cdot 2 \]

### Step 4: Solve for n
Rearranging the equation:
\[ 31 - 1 = (n - 1) \cdot 2 \]
\[ 30 = (n - 1) \cdot 2 \]
Dividing both sides by 2:
\[ 15 = n - 1 \]
Adding 1 to both sides:
\[ n = 16 \]

### Step 5: Use the sum formula for an arithmetic series
The sum (S) of an arithmetic series can be calculated using the formula:
\[ S = \frac{n}{2} \cdot (A + L) \]
Substituting the values we found:
\[ S = \frac{16}{2} \cdot (1 + 31) \]

### Step 6: Calculate the sum
Calculating the sum:
\[ S = 8 \cdot 32 \]
\[ S = 256 \]

### Final Answer
The sum of odd numbers from 1 to 31 is **256**.

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