Find the value of `1^(3) + 3^(3) + 5^(3) + 7^(3) + …….. + 29^(3)`

Options

A. 36100
B. 101025
C. 32500
D. 44700

Answer

To find the value of the series \(1^3 + 3^3 + 5^3 + 7^3 + \ldots + 29^3\), we can follow these steps:

### Step 1: Identify the series
The series consists of the cubes of odd natural numbers starting from 1 up to 29. The odd numbers in this range are:
\[ 1, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29 \]

### Step 2: Count the number of terms
The odd numbers from 1 to 29 can be expressed as:
\[ a_n = 2n - 1 \]
To find the number of terms, we set \(2n - 1 = 29\):
\[ 2n = 30 \\ n = 15 \]
So, there are 15 terms in the series.

### Step 3: Use the formula for the sum of cubes of the first \(n\) odd numbers
The formula for the sum of the cubes of the first \(n\) odd numbers is:
\[ \text{Sum} = n^2 \times (2n^2 - 1) \]
Substituting \(n = 15\):
\[ \text{Sum} = 15^2 \times (2 \times 15^2 - 1) \]

### Step 4: Calculate \(15^2\)
\[ 15^2 = 225 \]

### Step 5: Calculate \(2 \times 15^2 - 1\)
\[ 2 \times 15^2 = 2 \times 225 = 450 \\ 2 \times 15^2 - 1 = 450 - 1 = 449 \]

### Step 6: Calculate the total sum
Now we can substitute back into the formula:
\[ \text{Sum} = 225 \times 449 \]

### Step 7: Perform the multiplication
To calculate \(225 \times 449\), we can break it down:
\[ 225 \times 449 = 225 \times (450 - 1) = 225 \times 450 - 225 \]
Calculating \(225 \times 450\):
\[ 225 \times 450 = 101250 \\ 225 \times 449 = 101250 - 225 = 101025 \]

### Final Answer
Thus, the value of \(1^3 + 3^3 + 5^3 + 7^3 + \ldots + 29^3\) is:
\[ \boxed{101025} \]