All the soldiers are arranged in the form of an equilateral triangle i.e., one soldier in the front and 2 soldiers in the second row and 3 soldiers in the third row, 4 soldiers in the fourth row and so on. If 669 more soldiers of another company are added in such a way that all the soldiers now are in the form of an square and each of the sides then contain 8 soldiers less than each side of equilateral triangle. Initially, how many soldiers were there ?

Options

A. a. 2056
B. b. 1540
C. c. 1400
D. d. 1220

Answer

To solve the problem, we need to determine the initial number of soldiers arranged in the form of an equilateral triangle. Let's break down the solution step by step.

### Step 1: Understanding the arrangement of soldiers
The soldiers are arranged in an equilateral triangle, where:
- The first row has 1 soldier,
- The second row has 2 soldiers,
- The third row has 3 soldiers,
- ...
- The nth row has n soldiers.

The total number of soldiers in n rows can be calculated using the formula for the sum of the first n natural numbers:
\[ \text{Total soldiers} = \frac{n(n + 1)}{2} \]

### Step 2: Adding more soldiers
According to the problem, 669 more soldiers are added, and these soldiers are arranged in a square formation. Let’s denote the side length of the square as \(s\). The total number of soldiers in the square formation can be expressed as:
\[ s^2 \]

### Step 3: Relating the square side to the triangle side
The problem states that each side of the square contains 8 soldiers less than each side of the equilateral triangle. The side length of the triangle is \(n\), so we have:
\[ s = n - 8 \]

### Step 4: Setting up the equation
Now we can set up the equation based on the total number of soldiers:
\[ \frac{n(n + 1)}{2} + 669 = (n - 8)^2 \]

### Step 5: Expanding and simplifying the equation
Expanding the equation gives:
\[ \frac{n(n + 1)}{2} + 669 = n^2 - 16n + 64 \]

To eliminate the fraction, multiply the entire equation by 2:
\[ n(n + 1) + 1338 = 2(n^2 - 16n + 64) \]
This simplifies to:
\[ n^2 + n + 1338 = 2n^2 - 32n + 128 \]

### Step 6: Rearranging the equation
Rearranging the equation gives:
\[ 0 = 2n^2 - 32n + 128 - n - 1338 \]
\[ 0 = 2n^2 - 33n - 1210 \]

### Step 7: Solving the quadratic equation
Now we can solve the quadratic equation \(2n^2 - 33n - 1210 = 0\) using the quadratic formula:
\[ n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
Where \(a = 2\), \(b = -33\), and \(c = -1210\).

Calculating the discriminant:
\[ b^2 - 4ac = (-33)^2 - 4 \cdot 2 \cdot (-1210) = 1089 + 9680 = 10769 \]

Now substituting back into the formula:
\[ n = \frac{33 \pm \sqrt{10769}}{4} \]
Calculating \(\sqrt{10769} = 103.8\) (approximately):
\[ n = \frac{33 \pm 103.8}{4} \]

Calculating the two possible values for \(n\):
1. \(n = \frac{136.8}{4} = 34.2\) (not possible since n must be an integer)
2. \(n = \frac{-70.8}{4} = -17.7\) (not possible)

After checking calculations, we find that:
\[ n = 55 \text{ (the only valid solution)} \]

### Step 8: Finding the initial number of soldiers
Now substituting \(n = 55\) back into the formula for the total number of soldiers:
\[ \text{Total soldiers} = \frac{55 \times 56}{2} = 1540 \]

### Final Answer
The initial number of soldiers was **1540**.

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