Answer :
To solve the equation [tex]\( 2^3 + 4^3 + 6^3 + \cdots + (2n)^3 = 2n^2 (n+1)^2 \)[/tex], let's analyze and simplify both sides step-by-step.
### Step 1: Express the Left-Hand Side (LHS)
The left-hand side of the equation is the sum of cubes of even numbers from [tex]\(2\)[/tex] to [tex]\(2n\)[/tex]:
[tex]\[ 2^3 + 4^3 + 6^3 + \cdots + (2n)^3 \][/tex]
We can write the [tex]\(k\)[/tex]-th term in the series as [tex]\((2k)^3\)[/tex], where [tex]\( k \)[/tex] goes from 1 to [tex]\( n \)[/tex]. Therefore, the sum can be expressed as:
[tex]\[ \sum_{k=1}^n (2k)^3 \][/tex]
### Step 2: Simplify Each Term in the Sum
Notice that:
[tex]\[ (2k)^3 = 8k^3 \][/tex]
Thus, the sum becomes:
[tex]\[ \sum_{k=1}^n 8k^3 = 8 \sum_{k=1}^n k^3 \][/tex]
### Step 3: Use the Formula for the Sum of Cubes
The formula for the sum of the cubes of the first [tex]\( n \)[/tex] natural numbers is:
[tex]\[ \sum_{k=1}^n k^3 = \left( \frac{n(n+1)}{2} \right)^2 \][/tex]
### Step 4: Substitute the Sum of Cubes Formula
Substitute [tex]\(\left( \frac{n(n+1)}{2} \right)^2\)[/tex] into our earlier expression for the LHS:
[tex]\[ 8 \sum_{k=1}^n k^3 = 8 \left( \frac{n(n+1)}{2} \right)^2 \][/tex]
### Step 5: Simplify the Expression
Simplify the expression inside the parentheses and then the resulting expression:
[tex]\[ 8 \left( \frac{n(n+1)}{2} \right)^2 = 8 \left( \frac{n(n+1)}{2} \cdot \frac{n(n+1)}{2} \right) = 8 \cdot \frac{n^2(n+1)^2}{4} = 2n^2(n+1)^2 \][/tex]
### Step 6: Compare Both Sides
We have simplified the LHS to be:
[tex]\[ 2n^2(n+1)^2 \][/tex]
This matches exactly with the right-hand side of the given equation.
### Conclusion
Therefore, the original equation:
[tex]\[ 2^3 + 4^3 + 6^3 + \cdots + (2n)^3 = 2n^2 (n+1)^2 \][/tex]
is correct and has been verified step-by-step.
### Step 1: Express the Left-Hand Side (LHS)
The left-hand side of the equation is the sum of cubes of even numbers from [tex]\(2\)[/tex] to [tex]\(2n\)[/tex]:
[tex]\[ 2^3 + 4^3 + 6^3 + \cdots + (2n)^3 \][/tex]
We can write the [tex]\(k\)[/tex]-th term in the series as [tex]\((2k)^3\)[/tex], where [tex]\( k \)[/tex] goes from 1 to [tex]\( n \)[/tex]. Therefore, the sum can be expressed as:
[tex]\[ \sum_{k=1}^n (2k)^3 \][/tex]
### Step 2: Simplify Each Term in the Sum
Notice that:
[tex]\[ (2k)^3 = 8k^3 \][/tex]
Thus, the sum becomes:
[tex]\[ \sum_{k=1}^n 8k^3 = 8 \sum_{k=1}^n k^3 \][/tex]
### Step 3: Use the Formula for the Sum of Cubes
The formula for the sum of the cubes of the first [tex]\( n \)[/tex] natural numbers is:
[tex]\[ \sum_{k=1}^n k^3 = \left( \frac{n(n+1)}{2} \right)^2 \][/tex]
### Step 4: Substitute the Sum of Cubes Formula
Substitute [tex]\(\left( \frac{n(n+1)}{2} \right)^2\)[/tex] into our earlier expression for the LHS:
[tex]\[ 8 \sum_{k=1}^n k^3 = 8 \left( \frac{n(n+1)}{2} \right)^2 \][/tex]
### Step 5: Simplify the Expression
Simplify the expression inside the parentheses and then the resulting expression:
[tex]\[ 8 \left( \frac{n(n+1)}{2} \right)^2 = 8 \left( \frac{n(n+1)}{2} \cdot \frac{n(n+1)}{2} \right) = 8 \cdot \frac{n^2(n+1)^2}{4} = 2n^2(n+1)^2 \][/tex]
### Step 6: Compare Both Sides
We have simplified the LHS to be:
[tex]\[ 2n^2(n+1)^2 \][/tex]
This matches exactly with the right-hand side of the given equation.
### Conclusion
Therefore, the original equation:
[tex]\[ 2^3 + 4^3 + 6^3 + \cdots + (2n)^3 = 2n^2 (n+1)^2 \][/tex]
is correct and has been verified step-by-step.
