What is the value of `14^3+16^3+18^3+……+30^3`

Options

A. a) 134576
B. b) 120212
C. c) 115624
D. d) 111672

Answer

To find the value of \( 14^3 + 16^3 + 18^3 + \ldots + 30^3 \), we can follow these steps:

### Step 1: Identify the Sequence
The given series is \( 14^3, 16^3, 18^3, \ldots, 30^3 \). This is a series of cubes of even numbers starting from 14 to 30.

### Step 2: Rewrite the Terms
Notice that the terms can be expressed as:
\[ 14^3 = (2 \cdot 7)^3, \quad 16^3 = (2 \cdot 8)^3, \quad 18^3 = (2 \cdot 9)^3, \ldots, \quad 30^3 = (2 \cdot 15)^3 \]
This allows us to factor out \( 2^3 \) from each term:
\[ = 2^3 (7^3 + 8^3 + 9^3 + \ldots + 15^3) \]

### Step 3: Sum the Cubes
Now we need to find the sum \( 7^3 + 8^3 + 9^3 + \ldots + 15^3 \). We can use the formula for the sum of cubes of the first \( n \) natural numbers:
\[ \left( \frac{n(n+1)}{2} \right)^2 \]
However, we need the sum from 7 to 15, so we can calculate:
\[ \text{Sum from 1 to 15} - \text{Sum from 1 to 6} \]

### Step 4: Calculate the Sums
1. **Sum from 1 to 15**:
\[ S_{15} = \left( \frac{15 \cdot 16}{2} \right)^2 = (120)^2 = 14400 \]

2. **Sum from 1 to 6**:
\[ S_{6} = \left( \frac{6 \cdot 7}{2} \right)^2 = (21)^2 = 441 \]

### Step 5: Find the Required Sum
Now, we can find the sum from 7 to 15:
\[ S_{7 \text{ to } 15} = S_{15} - S_{6} = 14400 - 441 = 13959 \]

### Step 6: Multiply by \( 2^3 \)
Now, substituting back into our equation:
\[ 2^3 (7^3 + 8^3 + 9^3 + \ldots + 15^3) = 8 \cdot 13959 = 111672 \]

### Final Answer
Thus, the value of \( 14^3 + 16^3 + 18^3 + \ldots + 30^3 \) is:
\[ \boxed{111672} \]