Find out the unit digit in the following expression -
`1^3+2^3+3^3+4^3....+99^3`

Options

A. 0
B. 1
C. 2
D. 5

Answer

To find the unit digit of the expression \(1^3 + 2^3 + 3^3 + \ldots + 99^3\), we can follow these steps:

### Step 1: Understand the formula for the sum of cubes
The sum of the cubes of the first \(n\) natural numbers can be expressed using the formula:
\[ \left( \frac{n(n + 1)}{2} \right)^2 \]
This means that the sum of cubes from \(1^3\) to \(n^3\) is equal to the square of the sum of the first \(n\) natural numbers.

### Step 2: Identify \(n\)
In our case, we need to find the sum of cubes from \(1\) to \(99\). Therefore, we set \(n = 99\).

### Step 3: Apply the formula
Using the formula, we calculate:
\[ \text{Sum} = \left( \frac{99 \times (99 + 1)}{2} \right)^2 \]
Calculating \(99 + 1\):
\[ 99 + 1 = 100 \]
Now substituting back into the formula:
\[ \text{Sum} = \left( \frac{99 \times 100}{2} \right)^2 \]

### Step 4: Simplify the expression
Calculating \(99 \times 100\):
\[ 99 \times 100 = 9900 \]
Now divide by \(2\):
\[ \frac{9900}{2} = 4950 \]
Now we square \(4950\):
\[ \text{Sum} = 4950^2 \]

### Step 5: Find the unit digit
To find the unit digit of \(4950^2\), we only need to consider the unit digit of \(4950\), which is \(0\).

When squaring a number that ends in \(0\):
\[ 0^2 = 0 \]
Thus, the unit digit of \(4950^2\) is also \(0\).

### Final Answer
Therefore, the unit digit of the expression \(1^3 + 2^3 + 3^3 + \ldots + 99^3\) is \(0\).

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