If set N contains only consecutive positive integers, what

Question

If set N contains only consecutive positive integers, what is the sum of the numbers in set N ?

(1) Nineteen times the sum of the first number in the set and the last number in the set is 1729 
(2) There are 38 numbers in the set.

jlgdr · Accepted Answer

I think i'll go with (C) on this one. Please allow me to explain. So they are asking for the sum of the set of consecutive integers. We therefore need 2 things: 1) # of terms = Last - First + 1 2) Average of terms = (First + Last) / 2 Statement 1: We are given that 19 (F+L) = 1729 Then we can find F+L = 91. Hence, we can find the average. Now can we find Last - First? I'm afraid this is not possible. Statement 2: We are given the number of terms but we know nothing about the average of the terms. Hence Insuff (1) and (2): Now we have everything to find the sum of the set. # of terms from Statement 2 Average from Statement 1. Voila Hence answer is (C) Kudos if it helps!

sudhagar · Answer

I will go with C

Stmt 1) Let f be the first number and l be the last number 
19 ( f + l ) = 1729
-> f + l = 91 (A)

Hence INSUFF

Stmt 2) is INSUFF. 

Combining, we can calculate 'f' from (A) and Stmt 2 - Total 38 numbers. Hence suff.

Pls let me know if I have missed anything.

stne · Answer

I was able to get the answer using A only , please tell me what I have done wrong

sum of n consecutive terms =( n/2)( F+L), where n is the number of terms , f = first term and L is the last term 
now can 19 be expressed as n/2 ? of course we can , so we can write 38/2(f+L) = 1729 --> 19(F+L)=1729
this is the exact expression for the sum of n consecutive integers , hence the number of terms is 38 and the sum is 1729

Please let me know if I have erred somewhere.

statement 1 : 19(F+L)=1729 --> (38/2) ( F+L) - -> (n/2)( F+L) = 1729

Bunuel · Answer

We need to find the sum of N consecutive integers. The sum =  \frac{(first + last)}{2}*N .

From (1) we have that  19*(first + last) = 1729  -->  (first + last) = 91 . So, we need to find the value of  \frac{(first + last)}{2}*N = 91*\frac{N}{2} . How can you find the value of this expression? You cannot! for example, N can be 2 (45, 46) and in this case the sum is 91, or N can be 90 (1, 2, ..., 90) and in this case the sum is 4095.

Hope it helps.

stne · Answer

Thank you for your reply.

I could be wrong again, most probably am but I am just trying to see the light and find out what is wrong with my logic

suppose I was told that (n/2)(F+L) = 1729 where n is the number of terms in a consecutive series and f is the first term and l is the last term , Then I could immediately say that the sum of these n consecutive terms is 1729, could I not?

now aren't we give the exact same thing indirectly? We are told 19(F+L)= 1729 which is the same as saying 
(38/2)(F+L)= 1729 so looking at this and comparing with (n/2)(F+L)= 1729

why cannot I say n=38?

Bunuel · Answer

Suppose I tell you that the sum of the first and the last terms of n positive consecutive integers is 5. Can you find the sum of n consecutive integers?

The answer is NO. The set can be {2, 3} or {1, 2, 3, 4}. In the first case the sum is 5 and in the second case the sum is 10.

stne · Answer

I agree , sum of first and last term is not enough to give the number of consecutive terms in a sequence 
but we are given more than that, we are given 19 * ( F +T) this immediately forced me to tell that there are 38 consecutive terms in the sequence .

because 19(F+T) can we written as 38/2(F+T) which can be compared to n/2(F+T) I know 19 can we written in other forms as well but our aim is to get it in the form n/2 where n is a integer.

If I was given 20 *(F+T) and we were dealing with consecutive terms then could I not tell there are 40 terms in the sequence, by changing eqn. to 40/2(F+L)?

So F+T = 1729/19 if we look at this form of the equation , then certainly this seems insufficient
but after algebraic manipulation we get the equation in the form 19(F+T) = 1729 this form seems sufficient

I have learnt that in algebraic manipulation the same equation in certain form can give the answer well as in other forms it may not give the answer.

F+L = 1729/19 = insufficient but 19( F+T) = 1729 or (38/2) (F+L) = 1729 or (n/2)(F+L) is sufficient.

Is there a possibility that this could be a case here? Is there an issue of algebraic manipulation

Thank you for trying to make me understand.

Bunuel · Answer

Again from (38/2) (F+L) = 1729 you cannot tell that n=38.

anjancgc · Answer

Stne,

for the set {2,3} and {1,2,3,4} , if i were given 2* (F+T) = 10, as per you logic i can conclude there are 4 elements in the set. But that is not true as {2,3} can also be a set that satisfies 2* (F+T) = 10. The sum of {2,3} and {1,2,3,4} are different.

stne · Answer

Makes sense, you have managed to convince me.That's a wonderful first post by the way. +1 to you and also to Bunuel for his consistent assistance.

LogicGuru1 · Answer

Statment 1) 19 (First+last)=1729 First + Last=1729/19=91 We don't know first and last digit InSufficient (2) There are 38 numbers in the set Numbers can be anything (2,4,6,8,........... 38^{th} term) or (120,122,124,.......38^{th} term) Insufficient Merging both SUFFICIENT Here is the trick, most people don't know that If the total number of terms in a set containing even numbers is even then average of first and last term of set gives the average of the entire set. For example 2,4,6,8 standard way to find average= (2+4+6+8)/4=20/4= 5 The short cut method to find average is =(first+last)/2 ==> (2+8)/2= 5 This relationship hold true in case of even numbers having even terms

So we now know that 19(f+l)=1729 f+l=91 (f+l) =91/2=40.5 thats tha average To find the sum ;multiply it with the number of terms 38 * 40.5 = 1539 C is the answer

bumpbot · Answer

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If set N contains only consecutive positive integers, what

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