Problem Solving

I found this question-n-answer on net and somehow not convinced with the question -- seems question was not put "accurately", not the way it was solved.

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Ques: The sum of the even numbers between 1 and n is 79*80, where n is an odd number, then n=?

Sol: First term a=2, common difference d=2 since even number
an = a1 + (n - 1)d, n = 1, 2, ...
therefore sum to first n numbers of Arithmetic progression would be
n/2(2a+(n-1)d)
= n/2(2*2+(n-1)*2)=n(n+1) and this is equal to 79*80
therefore n=79 which is odd...
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In the question, it says "n is an odd number" and so it's pointing that n is the value of nth term and not the value of n (i.e. # of terms in the series). However, in solution, it used "n" as the # of terms in the series -- which is wrong.

As I understand,
Series would be: 2, 4, 6, 8 ..... (n-1)
getting # of terms (x) in the series:
n-1 = 2 + (x-1)* 2 => n-1 = 2 + 2x - 2 = 2x ==> x = (n-1)/2
therefore Sum = x/2 [ 2*2 + (x-1)*2] = (n-1)/4 [ 4 + 2x - 2] = (n-1)/4 [2 + 2x] = (n-1)/2 [x+1]
= (n-1)/2 [ (n-1)/2 +1 ] = (n-1)2 [ (n+1)/2] = ( n square - 1 ) / 4 = 79*80
==> n square = 79*80*4 +1 ==> n = 159 which is odd number

Can someone verify and reconfirm? Thanks

Yep, you did it correctly. That's one of the issues with doing problems from sets that float around the Internet.

gmatwarroom wrote:I found this question-n-answer on net and somehow not convinced with the question -- seems question was not put "accurately", not the way it was solved.

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Ques: The sum of the even numbers between 1 and n is 79*80, where n is an odd number, then n=?

Can someone verify and reconfirm? Thanks

I too get an answer of 159.

This problem becomes less tricky if we realize that 79*80 represents the formula for the sum of evenly spaced integers:

Sum = (number of integers) * (average of the biggest and smallest)

Thus, 79 = the number of even integers and 80 = the average of the biggest and smallest.
Let x = the biggest integer.
The smallest integer = 2.
(x+2)/2 = 80.
x+2 = 160
x = 158.
Thus, 79*80 represents the sum of the even integers from 2 to 158.
Since n is the next biggest odd integer, n=159.

hi mitch

u have considered 80 as average because it is given that n is odd number, rigth?

if it is not given then will it be difficult to say whether 79 or 80 is the average?

pls reply

hi mitch

u have considered 80 as average because it is given that n is odd number, rigth?

if it is not given then will it be difficult to say whether 79 or 80 is the average?

pls reply

Hi Mitch,

thanks for the formula -- helps.

However, even I have the same question that Krishna asked -- pls let us know.

Secondly, going by the formula, if 79 is the # of terms in the series then the greatest value (in the series) is straight to find i.e. 79*2 = 158 (as we are dealing with consecutive even numbers) - so, I don't see the need of using any further calculation of average etc.

What you think?