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If the sum of n (n>1) consecutive positive integers is 75. What is the  [#permalink]

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Question Stats: 31% (02:57) correct 69% (02:41) wrong based on 153 sessions

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If the sum of n(n>1) consecutive positive integers is 75. What is the sum of all possible values of n?

1. 5
2. 10
3. 16
4. 26
5. 50

Originally posted by nitingmat2018 on 21 May 2018, 09:51.
Last edited by pushpitkc on 21 May 2018, 12:56, edited 2 times in total.
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Re: If the sum of n (n>1) consecutive positive integers is 75. What is the  [#permalink]

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I just "see" 4 different variations. Therefore I would go with A, but I am missing a proper explanation. Pls help.
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If the sum of n (n>1) consecutive positive integers is 75. What is the  [#permalink]

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nitingmat2018 wrote:
If the sum of n (n>1) consecutive positive integers is 75. What is the sum of all possible values of n?

1. 5
2. 10
3. 16
4. 26
5. 50

75/2=37.5 median
2 terms are 37,38
with 3 terms, median will be 25,
so terms are 24,25,26
for 5 terms, median will be 15,
so terms are 13,14,15,16,17
6 terms are 10,11,12,13,14,15
2+3+5+6=16=sum of all possible values of n
---------------------------------------
oops! missed the 10 term sequence
thank you, pushpitkc

Originally posted by gracie on 21 May 2018, 12:46.
Last edited by gracie on 21 May 2018, 13:14, edited 2 times in total.
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If the sum of n (n>1) consecutive positive integers is 75. What is the  [#permalink]

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nitingmat2018 wrote:
If the sum of n(n>1) consecutive positive integers is 75. What is the sum of all possible values of n?

1. 5
2. 10
3. 16
4. 26
5. 50

n=2 | Terms are 37,38
n=3 | Terms are 24,25,26
n=5 | Terms are 13,14,15,16,17
n=6 | Terms are 10,11,12,13,14,15
n=10 | Terms are 3,4,5,6,7,8,9,10,11,12

Therefore, the sum of all possible values of n are 2+3+5+6+10 = 26(Option D)
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Re: If the sum of n (n>1) consecutive positive integers is 75. What is the  [#permalink]

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I like "pushpitkc" approach

n=2 | Terms are 37,38
n=3 | Terms are 24,25,26
n=5 | Terms are 13,14,15,16,17
n=6 | Terms are 10,11,12,13,14,15
n=10 | Terms are 3,4,5,6,7,8,9,10,11,12

Therefore, the sum of all possible values of n are 2+3+5+6+10 = 26(Option D)

But is There any other way we can deduce the same without making cases

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Re: If the sum of n (n>1) consecutive positive integers is 75. What is the  [#permalink]

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nitingmat2018 wrote:
If the sum of n(n>1) consecutive positive integers is 75. What is the sum of all possible values of n?

1. 5
2. 10
3. 16
4. 26
5. 50

Formula for sum of arithmetic progression is:
S = (a(1) +a(n)/2)*n = (2a(1)+(n-1)d)/2 * n
d = 1 (consecutive numbers)
S = (2a(1)+n-1)/2 * n = 75
So (2a(1)+n-1)n = 150
a(1) - positive and n is divisable by 150.
By checking we can see that fit next values for n: 2, 3, 5, 6, 10

Sum of these values - 26 - answer D
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Re: If the sum of n (n>1) consecutive positive integers is 75. What is the  [#permalink]

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nitingmat2018 wrote:
If the sum of n(n>1) consecutive positive integers is 75. What is the sum of all possible values of n?

1. 5
2. 10
3. 16
4. 26
5. 50

Sum of an $$AP = (n/2)[2a+(n-1)d]$$

Now, we have -

$$n (2a+n-1) = 25*6$$

So, n can be 2,3,5,6,10.
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Re: If the sum of n (n>1) consecutive positive integers is 75. What is the  [#permalink]

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1
I would solve it this way:

Since we need sum of n positive consecutive number to get 75, I can write it as:

n*k + n(n+1)/2 = 75

If I start putting n =1,2,3..., I should get k as integer.

K in integer for n = 2,3,5,6 and 10

So, the sum is 26
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Re: If the sum of n (n>1) consecutive positive integers is 75. What is the  [#permalink]

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nitingmat2018 wrote:
If the sum of n(n>1) consecutive positive integers is 75. What is the sum of all possible values of n?

1. 5
2. 10
3. 16
4. 26
5. 50

In my opinion,
to find the number of available sequences, we have to remember 2 possibilities:
first is ascending consecutive positive integers (where d = 1),
second is descending consecutive positive integers (where d = -1).

In this case, we have 2 equations:
first(when d = 1): 150 = n(2a+n-1)
second (when d = -1): 150 = n(2a-n+1)

from each equation, n = 2+3+5+6+10 = 26
so the sum = 26*2 = 52

which is not one of the mentioned answers in the question.
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