The sum of the first $n$ natural numbers is 55. What is $n$?
The sum of the first $n$ natural numbers is 55. What is $n$?
Show Hint
- 8
- 9
- 10
11
The Correct Option is C
Solution and Explanation
- Step 1: Formula for sum of first $n$ natural numbers - \[ S_n = \frac{n(n+1)}{2} \]
- Step 2: Substitute the given sum - \[ \frac{n(n+1)}{2} = 55 \]
- Step 3: Multiply through by 2 - \[ n(n+1) = 110 \]
- Step 4: Solve the quadratic - \[ n^2 + n - 110 = 0 \] Discriminant: $\Delta = 1 + 440 = 441$, $\sqrt{\Delta} = 21$.
- Step 5: Roots - \[ n = \frac{-1 \pm 21}{2} \] Only positive root is: \[ n = \frac{-1 + 21}{2} = \frac{20}{2} = 10 \]
- Step 6: Conclusion - $n = 10$, matching option (3).
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