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Question

# Find the number of ways in which 128 can be written as a sum of two or more consecutive natural numbers.

A

>2

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B

1

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C

2

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D

none of these

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Solution

## The correct option is D none of these The number of ways of expressing a number as a sum of two or more consecutive numbers = (No. of odd factors of that number -1) Since 128 is a power of 2, it does not have any odd factor. This means that it cannot be written as a sum of two or more consecutive natural numbers. Alternate Method: Sum of the consecutive numbers = 128 ⟹n2(2a+n−1)=128 (∵d=1) ⟹n(2a+n−1)=256 (where n>2) Now, all factors of 256 are even except 1 (n=1 is not possible). If n is even, then 'a' will never have an integral value. Hence, no such series of consecutive numbers is possible.

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