Question

The sum of the first n numbers, is 1275. The value of n is equal to

• A. 50
• B. 45
• C. 55
• D. 60

Hint:

To solve $n(n+1) = X$ quickly, take the square root of $X$. The integer part of the square root will usually be your $n$. \[ \sqrt{2550} \approx 50.4 \rightarrow n = 50 \]

Answer 1

The Correct Option is A

Concept: The sum of the first $n$ consecutive natural numbers ($1, 2, 3, \dots, n$) is a classic arithmetic progression. The formula to calculate this sum ($S_n$) is: \[ S_n = \frac{n(n+1)}{2} \]

Step 1: Set up the algebraic equation.
We are given that the sum $S_n$ is equal to 1275. We can substitute this into the formula: \[ 1275 = \frac{n(n+1)}{2} \]

Step 2: Solve the quadratic equation.
Multiply both sides by 2 to eliminate the denominator: \[ n(n+1) = 2 \times 1275 \] \[ n^2 + n = 2550 \] \[ n^2 + n - 2550 = 0 \]

Step 3: Finding the value of $n$.
We look for two consecutive numbers whose product is 2550. Let's test numbers near the square root of 2550 ($\sqrt{2550} \approx 50.49$): \[ 50 \times 51 = 2550 \] Therefore, the value of $n$ is 50. Final Answer: Option A