SOLUTION: 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?

Algebra.Com
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?
Answer by Edwin McCravy(20056)   (Show Source): You can put this solution on YOUR website!
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?
The formula for the sum of the first n natural numbers is 

Let the number missed be = k  

-k = 177

-k = 177

Multiply both sides by 2

nē + n - 2k = 354 

nē + n - 354 = 2k

k ≧ 1, so 2k ≧ 2

nē + n - 354 ≧ 2

nē + n - 352 ≧ 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

 +  - 177 = k

 +  - 177 = k

180.5 + 9.5 - 177 = k

13 = k

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

-------------------------------------------------

Checking:

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

1+2+3+4+5+6+7+8+9+10+11+12+14+15+16+17+18+19 = 177 

That checks.

Edwin
RELATED QUESTIONS

Out of 300 people sampled, 177 preferred Candidate A. Based on this, estimate what... (answered by Theo)
Adding together the first ten natural numbers is an example of a(n): a. series b.... (answered by jim_thompson5910)
Students were asked to write an algebraic expression for the sum of three consecutive... (answered by josgarithmetic)
If a natural number is written as the sum of three natural numbers, show that one of the... (answered by stanbon)
What is the sum of the first 50 natural... (answered by stanbon,checkley75)
What is the sum of the first 25 natural... (answered by richard1234)
what is the sum of the first 334 natural numbers? (answered by josmiceli)
find the sum of the first 328 even natural... (answered by Alan3354,solver91311)
the sum of three consecutive natural numbers if the first one is... (answered by ikleyn)