2015 Countdown Problem 12: Sums of consecutive numbers

How many integers between 2 and 2015 inclusive cannot be expressed as a sum of at least two consecutive positive integers?

This problem is part of the set 2015 Countdown Problems .


The answer is 10.

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3 solutions

Tijmen Veltman
Dec 22, 2014

The problem should really say "consecutive positive \text{positive} integers". Otherwise the answer would be 0, since k = n n + 1 k = n + 1 \sum_{k=-n}^{n+1}k=n+1 for any natural number n n .

True that, will amend accordingly. Thanks! :)

Wee Xian Bin - 6 years, 5 months ago

To add two or more consecutive numbers at least one has to be odd. Any power of 2 can not be built by summation that include an odd number. A n y o d d n u m b e r > 1 , can always be obtained by adding the following two consecutive numbers., T h e o d d n u m b e r 1 2 + T h e o d d n u m b e r + 1 2 . A n y e v e n n u m b e r may be obtain by adding only odd number of consecutive numbers, if the middle term is even. This middle term is also the average. So the sum would be number of terms (which is odd) * (middle term) 2 a n i n t e g e r \text{To add two or more consecutive numbers at least one has to be odd. Any}\\ \text{ power of 2 can not be built by summation that include an odd number.} \\~~\\ Any~\color{#3D99F6}{ odd~ number} > 1, \text{can always be obtained by adding the following }\\\text{two consecutive numbers.,}\\\dfrac{The~ odd ~number- 1} 2 +\dfrac{The~ odd ~number+ 1} 2.\\Any~\color{#D61F06}{ even~ number} \text{ may be obtain by adding only odd number of }\\ \text{consecutive numbers, if the middle term is even. This middle term is also} \\ \text{the average. So the sum would be } \\ \text{number of terms (which is odd) * (middle term)} \neq 2^{an~ integer}

Niranjan Khanderia - 5 years, 11 months ago
Shohag Hossen
Dec 9, 2014

2 (power) number's are only not consecutive : 2

4

8

16

32

64

128

256

512

1024

2048

4096 etc.

I don,t know why 2 (power) number's are not consecutive. But this is true.

From 2 to 2015 range our answer is 10.

answer = 10.

Moderator note:

This solution has been marked incomplete. You did not explain your reasoning.

Why is it that only numbers of form 2 n 2^n are not expressive as sum of consecutive integers ?

Venkata Karthik Bandaru - 6 years, 3 months ago

In response with respect to Challenge Master : Have you any explain that this solution is incorrect ?

Shohag Hossen - 6 years ago

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Shohag....look at Niranjan's solution... he was successful in explaining why only 2^n is a number that cant be represented as the sum of consecutive integers.

Anurag Singh - 5 years, 6 months ago

But 6 also cannot be expressed in that way

Rishik Jain - 6 years, 1 month ago

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1+2+3=6, so 6 is fine.

Doug Boyd - 6 years ago
Lew Sterling Jr
Dec 20, 2014

[2]

4

8

16

32

64

128

256

512

1024

[2015]

2048

Based on the intervals from 2 to 2015, or [2,2015], there are only 10 numbers.

Moderator note:

This solution has been marked incomplete. You did not explain your reasoning.

In response with respect to Challenge Master : Have you any explain that this solution is incorrect ?

Shohag Hossen - 6 years ago

I have also same concept

Shohag Hossen - 5 years, 6 months ago

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