The average of all odd numbers from 113 to 159 is_____

Options

A. 135
B. 134
C. 133
D. 136

Answer

To find the average of all odd numbers from 113 to 159, we can follow these steps:

### Step 1: Identify the odd numbers in the range
The first odd number in the range is 113, and the last odd number is 159. The odd numbers between these two values are:
113, 115, 117, ..., 159.

### Step 2: Determine the number of terms
To find the number of odd numbers, we can use the formula for the nth term of an arithmetic progression (AP):
- The first term (a) = 113
- The last term (l) = 159
- The common difference (d) = 2

The formula for the nth term of an AP is given by:
\[ l = a + (n-1) \cdot d \]
Rearranging this to find n:
\[ n = \frac{l - a}{d} + 1 \]
Substituting the values:
\[ n = \frac{159 - 113}{2} + 1 = \frac{46}{2} + 1 = 23 + 1 = 24 \]

### Step 3: Calculate the sum of the odd numbers
The sum of an arithmetic series can be calculated using the formula:
\[ S_n = \frac{n}{2} \cdot (a + l) \]
Substituting the values we found:
\[ S_{24} = \frac{24}{2} \cdot (113 + 159) = 12 \cdot 272 = 3264 \]

### Step 4: Calculate the average
The average (A) is given by the formula:
\[ A = \frac{S_n}{n} \]
Substituting the values:
\[ A = \frac{3264}{24} = 136 \]

Thus, the average of all odd numbers from 113 to 159 is **136**.

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