Question 823053
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?
<pre>
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

{{{(n^2+n)/2}}}-k = 177

Multiply both sides by 2

nē + n - 2k = 354 

nē + n - 354 = 2k

k &#8807; 1, so 2k &#8807; 2

nē + n - 354 &#8807; 2

nē + n - 352 &#8807; 0

By the quadratic formula, that has critical
numbers approximately 18.3 and -19.3

In integers the inequality has solutions 

n = -20,-21,-23,...  and n = 19, 20, 21,...
 
Since n is positive we take the first integer 
value exceeding 18.3, which is 19:

Substituting n=19 in

{{{n^2/2}}} + {{{n/2}}} - 177 = k

{{{19^2/2}}} + {{{19/2}}} - 177 = k

180.5 + 9.5 - 177 = k

13 = k

Answer:  the candidate added the integers 1 through 19, but omitted 13.

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Checking:

So since n = 19, the candidate added this with 13 omitted:

1+2+3+4+5+6+7+8+9+10+11+<font color="red">12+14</font>+15+16+17+18+19 = 177 

That checks.

Edwin</pre>