Mohan starts to calculate sum of all odd natural number less than 83. What result does he get ?
A. 1456
B. 1681
C. 1437
D. 1671

Options

A. C
B. A
C. B
D. D

Answer

To find the sum of all odd natural numbers less than 83, we can follow these steps:

### Step 1: Identify the odd natural numbers less than 83
The odd natural numbers less than 83 are: 1, 3, 5, 7, ..., 81.

### Step 2: Determine how many odd natural numbers are there
The largest odd number less than 83 is 81. The sequence of odd numbers can be expressed as:
1, 3, 5, ..., 81.

This is an arithmetic sequence where:
- The first term (a) = 1
- The last term (l) = 81
- The common difference (d) = 2

To find the number of terms (n) in this sequence, we can use the formula for the nth term of an arithmetic sequence:
\[ l = a + (n - 1) \cdot d \]
Substituting the known values:
\[ 81 = 1 + (n - 1) \cdot 2 \]
\[ 81 - 1 = (n - 1) \cdot 2 \]
\[ 80 = (n - 1) \cdot 2 \]
\[ n - 1 = 40 \]
\[ n = 41 \]

### Step 3: Use the formula for the sum of the first n odd natural numbers
The sum of the first n odd natural numbers is given by the formula:
\[ \text{Sum} = n^2 \]
Where n is the number of odd natural numbers.

Since we found that there are 41 odd natural numbers:
\[ \text{Sum} = 41^2 = 1681 \]

### Conclusion
Thus, the sum of all odd natural numbers less than 83 is **1681**.

### Final Answer
The result that Mohan gets is **B. 1681**.

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