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Is the sum of all the terms in a sequence of positive integers an odd

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Bunuel
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Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   23 May 2020, 07:13
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Is the sum of all the terms in a sequence of positive integers an odd integer?

(1) The sequence is composed of consecutive integers.
(2) There are 50 terms in the sequence.


DS21280
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   23 May 2020, 08:28
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Bunuel wrote:
Is the sum of all the terms in a sequence of positive integers an odd integer?

(1) The sequence is composed of consecutive integers.
(2) There are 50 terms in the sequence.


statement 1:
if the sequence has even number of consecutive integers then sum of these will be odd
if the sequence has odd number of consecutive integers then sum of these will be even

statement 2:
but we do not know how many of these 50 are even or odd integers.
if all are even or are odd then sum will be even.
if number of odd integers is odd then total sum will be odd.

combining both statements,
we know 25 are even and 25 are odd, thus sum will be odd.
Ans: C
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Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   08 Jun 2020, 09:31
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chetan2u, Bunuel, VeritasKarishma, nick1816, GMATBusters

Above question -- i tested values but could not understand the logic why 50 consecutives always give me an odd summation.

Do you think i got down the reasoning ? ( tried to summarize in the form of a take-away)

even number of consecutive integers

1) The sum may be even or odd
2) The number of evens WILL ALWAYS EQUAL number of odds.

Thus to quickly see if the sum is even or odd

-- Sum is even : if there are 2 evens and 2 odds or 4 evens and 4 odds or 10 evens and 10 odds.
-- Sum is odd : if there are 3 evens and 3 odds or 5 evens and 5 odds or 7 evens and 7 odds

odd number of consecutive integers

== the sum can be even or odd. The fail safe method to determine if this sum is odd or even w/o much calculation is the average {(H+L)/2}

--- if the average of the set is even == then the sum is also even
--- if the average of the set is odd ===then the sum is also odd
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   10 Jun 2020, 00:00
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jabhatta@umail.iu.edu wrote:
chetan2u, Bunuel, VeritasKarishma, nick1816, GMATBusters

Above question -- i tested values but could not understand the logic why 50 consecutives always give me an odd summation.

Do you think i got down the reasoning ? ( tried to summarize in the form of a take-away)

even number of consecutive integers

1) The sum may be even or odd
2) The number of evens WILL ALWAYS EQUAL number of odds.

Thus to quickly see if the sum is even or odd

-- Sum is even : if there are 2 evens and 2 odds or 4 evens and 4 odds or 10 evens and 10 odds.
-- Sum is odd : if there are 3 evens and 3 odds or 5 evens and 5 odds or 7 evens and 7 odds

odd number of consecutive integers

== the sum can be even or odd. The fail safe method to determine if this sum is odd or even w/o much calculation is the average {(H+L)/2}

--- if the average of the set is even == then the sum is also even
--- if the average of the set is odd ===then the sum is also odd


Just see how many odd integers you have.
If you have even number of odd integers, sum is even (all the 1s get paired).
If you have odd number of odd integers, sum is odd (one 1 remains extra).
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   05 Oct 2021, 22:12
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Sidharth003 wrote:
VeritasKarishma wrote:
jabhatta@umail.iu.edu wrote:
chetan2u, Bunuel, VeritasKarishma, nick1816, GMATBusters

Above question -- i tested values but could not understand the logic why 50 consecutives always give me an odd summation.

Do you think i got down the reasoning ? ( tried to summarize in the form of a take-away)

even number of consecutive integers

1) The sum may be even or odd
2) The number of evens WILL ALWAYS EQUAL number of odds.

Thus to quickly see if the sum is even or odd

-- Sum is even : if there are 2 evens and 2 odds or 4 evens and 4 odds or 10 evens and 10 odds.
-- Sum is odd : if there are 3 evens and 3 odds or 5 evens and 5 odds or 7 evens and 7 odds

odd number of consecutive integers

== the sum can be even or odd. The fail safe method to determine if this sum is odd or even w/o much calculation is the average {(H+L)/2}

--- if the average of the set is even == then the sum is also even
--- if the average of the set is odd ===then the sum is also odd


Just see how many odd integers you have.
If you have even number of odd integers, sum is even (all the 1s get paired).
If you have odd number of odd integers, sum is odd (one 1 remains extra).


Hey VeritasKarishma had a follow up question to your post. Would it be fair to say:
a) Irrespective of whether we have an even number or an odd number of "even integers" i.e eoe(2 evens) or oeo(1 evens) we will always get even whereas like you mentioned
b) if odd number of odds=sum is odd and if even number of odds sum is even

And these principles apply irrespective of whether the set has consecutive or random integers right and irrespective of number of integers in the set? i.e all that matters is the number of odds(i.e only if odd number of odds sum is odd else whatever be the case always even)

Is this understanding correct? or am I missing some possible case?

Looking forward to hear from you
Thanks:)


Yes.
I think of even numbers as a collection of pairs of ones. 11 11 11 ...
and odd numbers as pairs of 1s with a loose 1 hanging around 11 11 11 11 1 ...

So when I add evens, any evens, they always give a collection of pairs of ones only.
When I add odds, it depends on how many odds I am adding. When I add an even number of odds, all hanging ones get partners to stick together to form a pair of ones. So I get again a collection of pairs of ones.
When I add an odd number of odds, there is one 1 that still hangs about. So I get an odd number.
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   11 Oct 2022, 00:17
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Where is it mentioned that the terms are equally skewed, having same number as positive as negative terms?
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   23 May 2020, 10:29
Statement (1) The sequence is composed of consecutive integers.
We know that:
1. The sum of even/odd number of consecutive odd/even integers is even.
2. The sum of odd number of consecutive integers is odd, but the sum of even number of consecutive integers is even.
So it could be anything.
Thus the statement is Insufficient.


Statement (2) There are 50 terms in the sequence.
Now, it could any 50 terms,
1 even 49 odd, => sum would be odd.
1 odd, 49 even => sum would be odd.
25 even 25 odd => odd
50 even or 50 odd = even.

Again we cannot say with certainty which one of the above could be the case.
So Insufficient.


Statement 1 + Statement 2: Consecutive positive integers and 50 terms in the sequence.
=> 25 even and 25 odd => even + odd => odd
Thus when we combine the 2 statements we get the sum as odd.
So Sufficient

Answer C.
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   08 Jun 2020, 09:50
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Take any 2 consecutive integers and add them, the sum of the pair is always odd ( because one will be odd and other will be even)

now 50 consecutive = 25 such pairs = 25* odd = odd

(50 consecutive = 25 even & 25 odd)


I hope it is clear.
jabhatta@umail.iu.edu wrote:
chetan2u, Bunuel, VeritasKarishma, nick1816, GMATBusters

Above question -- i tested values but could not understand the logic why 50 consecutives always give me an odd summation.

Do you think i got down the reasoning ? ( tried to summarize in the form of a take-away)

even number of consecutive integers

1) The sum may be even or odd
2) The number of evens WILL ALWAYS EQUAL number of odds.

Thus to quickly see if the sum is even or odd

-- Sum is even : if there are 2 evens and 2 odds or 4 evens and 4 odds or 10 evens and 10 odds.
-- Sum is odd : if there are 3 evens and 3 odds or 5 evens and 5 odds or 7 evens and 7 odds

odd number of consecutive integers

== the sum can be even or odd. The fail safe method to determine if this sum is odd or even w/o much calculation is the average {(H+L)/2}

--- if the average of the set is even == then the sum is also even
--- if the average of the set is odd ===then the sum is also odd
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   08 Jun 2020, 10:14
Bunuel wrote:
Is the sum of all the terms in a sequence of positive integers an odd integer?

(1) The sequence is composed of consecutive integers.
(2) There are 50 terms in the sequence.


DS21280


Asked: Is the sum of all the terms in a sequence of positive integers an odd integer?

(1) The sequence is composed of consecutive integers.
Since number of terms and initial terms are unknown
NOT SUFFICIENT

(2) There are 50 terms in the sequence.
Since there is no information about terms.
NOT SUFFICIENT

(1) + (2)
(1) The sequence is composed of consecutive integers.
(2) There are 50 terms in the sequence.
25 integers are even - > sum is even
25 integers are odd -> sum is odd
Sum of 50 consecutive integers - > even + odd - > odd
SUFFICIENT

IMO C
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   22 Jul 2020, 13:15
Bunuel - any similar questions you can think off ? i have done the questions seen in the similar topics but none of them were exactly like this question

thank you !
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   23 Jul 2020, 22:14
bm2201 wrote:
Statement (1) The sequence is composed of consecutive integers.
We know that:
1. The sum of even/odd number of consecutive odd/even integers is even.
2. The sum of odd number of consecutive integers is odd, but the sum of even number of consecutive integers is even.
So it could be anything.
Thus the statement is Insufficient.


Statement (2) There are 50 terms in the sequence.
Now, it could any 50 terms,
1 even 49 odd, => sum would be odd.
1 odd, 49 even => sum would be odd.
25 even 25 odd => odd
50 even or 50 odd = even.

Again we cannot say with certainty which one of the above could be the case.
So Insufficient.


Statement 1 + Statement 2: Consecutive positive integers and 50 terms in the sequence.
=> 25 even and 25 odd => even + odd => odd
Thus when we combine the 2 statements we get the sum as odd.
So Sufficient

Answer C.


If the sum of odd number of consecutive integers odd, then how come 1+2+3= 6?
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   25 Nov 2020, 12:48
Bunuel wrote:
Is the sum of all the terms in a sequence of positive integers an odd integer?

(1) The sequence is composed of consecutive integers.
(2) There are 50 terms in the sequence.


DS21280



(1) We don't know how many integers are in the set.

We could have 1 odd integer in the set, which would give us an odd sum.

We could have 1 even integer in the set, which would give us an even sum.

INSUFFICIENT.

(2) If there are 50 terms in the sequence, we could have 50 2's, which could give us a sum of 100.

We could have 49 1's and one 2, which would give us an odd sum (101)

INSUFFICIENT.

(1&2) If we have 50 consecutive integers, we will have 25 even integers and 25 odd integers.

Sum of 25 even integers = even
Sum of 25 odd integers = odd
even + odd = odd

SUFFICIENT.

Answer is C.
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink] New post   05 Oct 2021, 02:17
VeritasKarishma wrote:
jabhatta@umail.iu.edu wrote:
chetan2u, Bunuel, VeritasKarishma, nick1816, GMATBusters

Above question -- i tested values but could not understand the logic why 50 consecutives always give me an odd summation.

Do you think i got down the reasoning ? ( tried to summarize in the form of a take-away)

even number of consecutive integers

1) The sum may be even or odd
2) The number of evens WILL ALWAYS EQUAL number of odds.

Thus to quickly see if the sum is even or odd

-- Sum is even : if there are 2 evens and 2 odds or 4 evens and 4 odds or 10 evens and 10 odds.
-- Sum is odd : if there are 3 evens and 3 odds or 5 evens and 5 odds or 7 evens and 7 odds

odd number of consecutive integers

== the sum can be even or odd. The fail safe method to determine if this sum is odd or even w/o much calculation is the average {(H+L)/2}

--- if the average of the set is even == then the sum is also even
--- if the average of the set is odd ===then the sum is also odd


Just see how many odd integers you have.
If you have even number of odd integers, sum is even (all the 1s get paired).
If you have odd number of odd integers, sum is odd (one 1 remains extra).


Hey VeritasKarishma had a follow up question to your post. Would it be fair to say:
a) Irrespective of whether we have an even number or an odd number of "even integers" i.e eoe(2 evens) or oeo(1 evens) we will always get even whereas like you mentioned
b) if odd number of odds=sum is odd and if even number of odds sum is even

And these principles apply irrespective of whether the set has consecutive or random integers right and irrespective of number of integers in the set? i.e all that matters is the number of odds(i.e only if odd number of odds sum is odd else whatever be the case always even)

Is this understanding correct? or am I missing some possible case?

Looking forward to hear from you
Thanks:)
GMAT Club Bot
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Re: Is the sum of all the terms in a sequence of positive integers an odd [#permalink]
05 Oct 2021, 02:17
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