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If z1, z2, z3,..........,zn is a series of consecutive +ve integers, is the sum of all the integers in this series odd?
1) Average of all the numbers in the series is odd
2) n is odd
2) n is odd
- a)Exactly one of the statements can answer the question
- b)Both statements are required to answer the question
- c)Each statement can answer the question individually
- d)More information is required as the information provided is insufficient to answer the question
Correct answer is option 'A'. Can you explain this answer?
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If z1, z2, z3,..........,zn is a series of consecutive +ve integers, i...
Solution:
Given, z1, z2, z3,..........,zn is a series of consecutive integers.
To find: Is the sum of all the integers in this series odd?
Let's consider the sum of the series as S.
Statement 1: Average of all the numbers in the series is odd
If the average of the series is odd, then the sum of the series must also be odd.
As the series is consecutive integers,
Average of the series = (z1+zn)/2
Sum of the series = n*(z1+zn)/2
If the average is odd, then (z1+zn)/2 must be an odd integer.
Therefore, z1 and zn must have the same parity (both odd or both even).
So, sum of the series = n*(z1+zn)/2 = n*(odd integer) = odd integer
Hence, this statement alone is sufficient to answer the question.
Statement 2: n is odd
If n is odd, then there are odd number of integers in the series.
Let's consider a few examples:
- If n=3, the series would be z1, z2, z3 and the sum would be z1+z2+z3.
- If n=5, the series would be z1, z2, z3, z4, z5 and the sum would be z1+z2+z3+z4+z5.
In both cases, the sum of the series would be odd only if the middle term (z2 or z3 and z3 or z4 respectively) is odd. This is because the sum of the remaining terms on both sides of the middle term would be even.
Therefore, this statement alone is not sufficient to answer the question.
Hence, the answer is option A - Exactly one of the statements can answer the question.
Given, z1, z2, z3,..........,zn is a series of consecutive integers.
To find: Is the sum of all the integers in this series odd?
Let's consider the sum of the series as S.
Statement 1: Average of all the numbers in the series is odd
If the average of the series is odd, then the sum of the series must also be odd.
As the series is consecutive integers,
Average of the series = (z1+zn)/2
Sum of the series = n*(z1+zn)/2
If the average is odd, then (z1+zn)/2 must be an odd integer.
Therefore, z1 and zn must have the same parity (both odd or both even).
So, sum of the series = n*(z1+zn)/2 = n*(odd integer) = odd integer
Hence, this statement alone is sufficient to answer the question.
Statement 2: n is odd
If n is odd, then there are odd number of integers in the series.
Let's consider a few examples:
- If n=3, the series would be z1, z2, z3 and the sum would be z1+z2+z3.
- If n=5, the series would be z1, z2, z3, z4, z5 and the sum would be z1+z2+z3+z4+z5.
In both cases, the sum of the series would be odd only if the middle term (z2 or z3 and z3 or z4 respectively) is odd. This is because the sum of the remaining terms on both sides of the middle term would be even.
Therefore, this statement alone is not sufficient to answer the question.
Hence, the answer is option A - Exactly one of the statements can answer the question.
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If z1, z2, z3,..........,zn is a series of consecutive +ve integers, i...
Correct answer is option 'A'
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If z1, z2, z3,..........,zn is a series of consecutive +ve integers, is the sum of all the integers in this series odd?1) Average of all the numbers in the series is odd2) n is odd a)Exactly one of the statements can answer the questionb)Both statements are required to answer the questionc)Each statement can answer the question individuallyd)More information is required as the information provided is insufficient to answer the questionCorrect answer is option 'A'. Can you explain this answer?
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