The sum of n consecutive positive integers is 45. What is the value of n?

(1) n is even

(2) n < 9

OPEN DISCUSSION OF THIS QUESTION IS HERE: the-sum-of-n-consecutive-positive-integers-is-45-what-is-126861.html
_________________

Stay Hungry, Stay Foolish

My approach is:

taking them together:

let's say n =2, sum = 2m+1 = 45, so numbers are 22 and 23
n=6, numbers are 5,6,7,8,9,10

still insuff

The formula n(n+1)/2 is only valid if you have series starting from 1.
So it will be valid for 1+2+3+...+9 But it will not be valid for 4+5+6+7 or for that matter any series that does not start with 1.

The sum of n consecutive positive integers is 45. What is the value of n?

(1) n is even

(2) n < 9
_________________

Chinese Democracy is misunderstood...at your nearest BestBuy.

Best AWA guide here: how-to-get-6-0-awa-my-guide-64327.html

Let the series be 1,2,3,....,m,.......m+n

Now sum of all positive integers from m+1 to m+n = (m+n)(m+n+1)/2 - m(m+1)/2
= 45

After simplification Implies: n(2m+n+1)= 90

90 = 9X10 = 6X15 = 5X18 = 3X30 = 2X 45

Now accordingly we can find n & m by substituting values above with n= 9, 6,5,3,2 and then we get m= 0, 4, 6, 13, 21

Statement1: n is even : n can take values of 6 and 2 so insufficient

Statement 2: N<9 : n can take 9, 6, 5, 3, 2 hence insufficient

Bothe together: N can take 6 and 2 thus Ans is E

But how can you use your approach for slightly modified problem: "The sum of n consecutive positive integers is 47. What is the value of n?"
_________________

iOS/Android: GMAT ToolKit - The bestselling GMAT prep app | GMAT Club (free) | PrepGame | GRE ToolKit | LSAT ToolKit
PROMO: Are you an exiting GMAT ToolKit (iOS) user? Get GMAT ToolKit 2 (iOS) for free* (read more)
Math: GMAT Math Book ||| General: GMATTimer ||| Chicago Booth: Slide Presentation
The People Who Are Crazy Enough to Think They Can Change the World, Are the Ones Who Do.

Please, give me two examples with different n and that satisfy both conditions in the case of sum=47
_________________

iOS/Android: GMAT ToolKit - The bestselling GMAT prep app | GMAT Club (free) | PrepGame | GRE ToolKit | LSAT ToolKit
PROMO: Are you an exiting GMAT ToolKit (iOS) user? Get GMAT ToolKit 2 (iOS) for free* (read more)
Math: GMAT Math Book ||| General: GMATTimer ||| Chicago Booth: Slide Presentation
The People Who Are Crazy Enough to Think They Can Change the World, Are the Ones Who Do.

Maybe I've understood incorrectly, Alpha_plus_gamma's approach in based on S = N [2*a1 + (N-1)d] /2 formula and he chose E because too many unknowns. So, if we substitute 45 for 47 his approach is insensitive to this substitution, isn't it?
_________________

iOS/Android: GMAT ToolKit - The bestselling GMAT prep app | GMAT Club (free) | PrepGame | GRE ToolKit | LSAT ToolKit
PROMO: Are you an exiting GMAT ToolKit (iOS) user? Get GMAT ToolKit 2 (iOS) for free* (read more)
Math: GMAT Math Book ||| General: GMATTimer ||| Chicago Booth: Slide Presentation
The People Who Are Crazy Enough to Think They Can Change the World, Are the Ones Who Do.

When dealing with the sum of evenly spaced integers (also known as arithmetic progression or sequence), just remember the
simple, universal formula: Sum = Average* (# of values). [* represents the multiplication sign.] The space is 1 here.
Applied to this problem, Sum = n*(a1+aN)/2. a1 and aN are the first term and the last term of the sequence respectively, (a1+ aN)/2 being the average, but also the median or the midpoint. They all coincide when dealing with an arithmetic progression. n is of course the number of terms.
So, it follows that n(a1 + aN)/2 = 45 and (a1 + aN) = 90/n. Since a1 + a2 is an integer, so must be 90/n. It then boils down to finding the positive factors of 90. Let's find the prime factorization of 90. 90 = 9*10 = 3*3*2*5 = 3^2 * 2 * 5.
We can easily list all the positive factors of 90; there are 12 of them. I actually have a formula to find them fast but that's another topic.
1/Since n is even, let's focus on the even values of n; they are 2, 6, 10, 18, 30, and 90.
But if n = 10 then the median of the sequence will be (a1 + a2)/2 = 45/10 = 4.5 and some of the terms will be negative (NOT Allowed).
Any n > 10 will have to be rejected for the same reason. Therefore n must be 2 or 6: UNSUFF.
2/ n < 9 includes more than one n (2, 3, 5, 6): UNSUFF.
Together, we are still stuck with 2 and 6.
Note that when n = 2, the average is 45/2 = 22.5; then a1 = 22 and aN = a2 = 23. 22+23 = 45
When n = 6, then a1 = 5; a2 = 6; a3 =7; a4 = 8, a5 = 9, a6 = 10. 5+6+7 +8+9+10 = 45.
Regards,





_________________

Dakar Azu is The GMAT Doctor.
Dakar is an experienced GMAT teacher who can be reached at http://700gmatclub.com. He prepares aspiring business students thoroughly to get them well over the GMAT 700-score hurdle through his online GMAT courses.

to formulas are handy here..

sum=N/2 * {2a+(n-1)d} where d is the difference, in our case= 1, a=the first number in the series..we dont what that is..)

second formula..
n*(na+nd)/2=sum where na is the first number in the series and nd=last number in the series..i.e avg*number of terms=sum..

{na+nd}n=90..

1)n is even..

na+nd=90/n now..lets see what n can divide 90 evenly..n=2, 6..lets stick with that

if n=2..na+nd=45.. lets see what those numbers..23+22=45.ok great!
if n=6 then lets (2*a+n-1}=90
suppose a=5.. n{10+n-1}=n{n+9}=90..n=6..works!

you can also check via.. {na+nd}=15...if na=5 and Nd=10...works..

Insuff

2) N<9

well i just showed 2 examples of n=2 and 6...Insuff
E it is