Question

while adding the first few continuous natural numbers ,a candidate missed one of the numbers and wrote the answer as 177.what was the number missed? ​

Answer 1

The formula for the sum of the first n natural numbers is n(n+1)/2

Let the number missed be = k  

n(n+1)/2 -k = 177

Multiply both sides by 2

n2 + n - 2k = 354

n2 + n - 354 = 2k

k ≧ 1, so 2k ≧ 2

n2 + n - 354 ≧ 2

n2 + n - 352 ≧ 0

In integers the inequality has solutions

n = -20,-21,-23,...  and n = 19, 20, 21,...

Substituting n=19 in

n(n+1)/2 - 177 = k

19(19+1)/2 - 177 = k

190 - 177 = k

13 = k