If `1 + 2 + 3 + …+ k = N^2` and N is less than 100 then the value of k can be, where `N in N` Natural Numbers:

Options

A. a.8
B. b. 1 and 49
C. c.8 and 36
D. d. both (a) and (b)

Answer

To solve the problem where \(1 + 2 + 3 + \ldots + k = N^2\) and \(N < 100\), we can follow these steps:

### Step 1: Understand the formula for the sum of the first \(k\) natural numbers
The sum of the first \(k\) natural numbers can be expressed using the formula:
\[ S_k = \frac{k(k + 1)}{2} \]
where \(S_k\) is the sum of the first \(k\) numbers.

### Step 2: Set up the equation based on the problem statement
According to the problem, we have:
\[ \frac{k(k + 1)}{2} = N^2 \]
This means that \(k(k + 1) = 2N^2\).

### Step 3: Analyze the equation
We need to find values of \(k\) such that \(k(k + 1)\) is twice a perfect square. Since \(N\) is a natural number and less than 100, \(N^2\) can take values from \(1^2\) to \(99^2\).

### Step 4: Calculate possible values of \(N^2\)
The maximum value of \(N^2\) when \(N < 100\) is:
\[ N^2 < 100^2 = 10000 \]
Thus, \(N^2\) can take values: \(1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625, 676, 729, 784, 841, 900, 961\).

### Step 5: Find corresponding values of \(k\)
We need to find \(k\) such that:
\[ k(k + 1) = 2N^2 \]
This means we can test values of \(N^2\) and see if \(k(k + 1)\) equals \(2N^2\).

### Step 6: Testing values
1. For \(N = 1\):
\[ 2N^2 = 2 \times 1^2 = 2 \implies k(k + 1) = 2 \implies k = 1 \]

2. For \(N = 2\):
\[ 2N^2 = 2 \times 2^2 = 8 \implies k(k + 1) = 8 \implies k = 2 \]

3. For \(N = 3\):
\[ 2N^2 = 2 \times 3^2 = 18 \implies k(k + 1) = 18 \implies k = 4 \]

4. For \(N = 6\):
\[ 2N^2 = 2 \times 6^2 = 72 \implies k(k + 1) = 72 \implies k = 8 \]

5. For \(N = 7\):
\[ 2N^2 = 2 \times 7^2 = 98 \implies k(k + 1) = 98 \implies k = 9 \]

6. For \(N = 8\):
\[ 2N^2 = 2 \times 8^2 = 128 \implies k(k + 1) = 128 \implies k = 11 \]

7. For \(N = 9\):
\[ 2N^2 = 2 \times 9^2 = 162 \implies k(k + 1) = 162 \implies k = 12 \]

8. For \(N = 35\):
\[ 2N^2 = 2 \times 35^2 = 2450 \implies k(k + 1) = 2450 \implies k = 49 \]

### Conclusion
The possible values of \(k\) that satisfy the condition \(1 + 2 + 3 + \ldots + k = N^2\) for \(N < 100\) are \(1, 8, 49\).