Discussion
Shantanu was given a task of adding a certain number of consecutive natural numbers starting from 1. By mistake, he missed a number during addition. He obtained the sum as 800. Find the number that he missed.
Answer: B
The missed number is 20.
The sum of the first \(n\) natural numbers is \(\frac{n(n+1)}{2}\). The full sum must be just above 800, because one number was left out:
\[ \begin{aligned} n = 39 &: \frac{39 \times 40}{2} = 780 \\ n = 40 &: \frac{40 \times 41}{2} = 820 \end{aligned} \]
780 is below 800, so Shantanu must have been adding 1 to 40, whose total is 820. The missing number is \(820 - 800 = 20\).
A larger \(n\) does not work: for \(n = 41\) the total is 861, and the missing number would be 61, which is larger than 41 and so cannot be one of the numbers. 15, 25 and 10 would leave sums of 805, 795 and 810, not 800.
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