The formula of the sum of first n natural numbers is S = n(n + 1)/2. If the sum of first n natural number is 325 then find n.

To find the value of n when the sum of the first n natural numbers equals 325, we can use the formula:

S = n(n + 1)/2

Setting the equation equal to 325 gives us:

Equation Setup

n(n + 1)/2 = 325

To eliminate the fraction, multiply both sides by 2:

Multiplying Through

n(n + 1) = 650

Rearranging the Equation

This can be rewritten as:

n² + n - 650 = 0

Solving the Quadratic Equation

Now, we can use the quadratic formula:

n = [-b ± √(b² - 4ac)] / 2a

Here, a = 1, b = 1, and c = -650.

Calculating the Discriminant

First, calculate the discriminant:

b² - 4ac = 1² - 4(1)(-650) = 1 + 2600 = 2601

Finding n

Now plug this back into the formula:

n = [-1 ± √2601] / 2

√2601 = 51, so:

n = [-1 ± 51] / 2

This gives us two potential solutions:

• n = (50) / 2 = 25
• n = (-52) / 2 = -26 (not valid since n must be positive)

Final Answer

The value of n is 25.