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Consider a sequence of twelve consecutive even integers. If the average of the first seven is n, then the average of all the twelve integers is:
- a)n
- b)n + 5
- c)K x n, where K is a function of n
- d)n + (5/7)
Correct answer is option 'B'. Can you explain this answer?
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Verified Answer
Consider a sequence of twelve consecutive even integers. If the averag...
Average of twelve consecutive even integers is the average of the sixth and the seventh even integers.
As the average of the first seven is n, the fourth even integer is n.
The sixth and the seventh integers are (n + 4) and (n + 6).
Average of (n + 4) and (n + 6) = (n + 5).
Hence, option 2.
As the average of the first seven is n, the fourth even integer is n.
The sixth and the seventh integers are (n + 4) and (n + 6).
Average of (n + 4) and (n + 6) = (n + 5).
Hence, option 2.
Most Upvoted Answer
Consider a sequence of twelve consecutive even integers. If the averag...
Problem:
Consider a sequence of twelve consecutive even integers. If the average of the first seven is n, then the average of all the twelve integers is:
a) n
b) n + 5
c) K * n, where K is a function of n
d) (5/7) * n
Solution:
To solve this problem, let's break it down into smaller steps.
Step 1: Express the first seven integers in terms of n
Let's assume the first even integer in the sequence is x. Since the sequence consists of consecutive even integers, the next six even integers can be expressed as x + 2, x + 4, x + 6, x + 8, x + 10, and x + 12.
The average of the first seven integers is given by:
average = (x + (x + 2) + (x + 4) + (x + 6) + (x + 8) + (x + 10) + (x + 12))/7
Simplifying the above expression, we get:
average = (7x + 42)/7
Since the average is given as n, we can equate the two expressions:
n = (7x + 42)/7
Cross-multiplying, we get:
7n = 7x + 42
Simplifying further, we get:
7n - 42 = 7x
x = (7n - 42)/7
x = n - 6
Therefore, the first seven integers in terms of n are:
n - 6, n - 4, n - 2, n, n + 2, n + 4, n + 6
Step 2: Find the average of all twelve integers
To find the average of all twelve integers, we need to consider the entire sequence. The remaining five even integers can be expressed as n + 8, n + 10, n + 12, n + 14, and n + 16.
The average of all twelve integers is given by:
average = (n - 6 + n - 4 + n - 2 + n + n + 2 + n + 4 + n + 6 + n + 8 + n + 10 + n + 12 + n + 14 + n + 16)/12
Simplifying the above expression, we get:
average = (12n + 96)/12
Simplifying further, we get:
average = n + 8
Therefore, the average of all twelve integers is n + 8.
Final Answer:
The average of all twelve integers is n + 8, which corresponds to option b) in the given choices.
Consider a sequence of twelve consecutive even integers. If the average of the first seven is n, then the average of all the twelve integers is:
a) n
b) n + 5
c) K * n, where K is a function of n
d) (5/7) * n
Solution:
To solve this problem, let's break it down into smaller steps.
Step 1: Express the first seven integers in terms of n
Let's assume the first even integer in the sequence is x. Since the sequence consists of consecutive even integers, the next six even integers can be expressed as x + 2, x + 4, x + 6, x + 8, x + 10, and x + 12.
The average of the first seven integers is given by:
average = (x + (x + 2) + (x + 4) + (x + 6) + (x + 8) + (x + 10) + (x + 12))/7
Simplifying the above expression, we get:
average = (7x + 42)/7
Since the average is given as n, we can equate the two expressions:
n = (7x + 42)/7
Cross-multiplying, we get:
7n = 7x + 42
Simplifying further, we get:
7n - 42 = 7x
x = (7n - 42)/7
x = n - 6
Therefore, the first seven integers in terms of n are:
n - 6, n - 4, n - 2, n, n + 2, n + 4, n + 6
Step 2: Find the average of all twelve integers
To find the average of all twelve integers, we need to consider the entire sequence. The remaining five even integers can be expressed as n + 8, n + 10, n + 12, n + 14, and n + 16.
The average of all twelve integers is given by:
average = (n - 6 + n - 4 + n - 2 + n + n + 2 + n + 4 + n + 6 + n + 8 + n + 10 + n + 12 + n + 14 + n + 16)/12
Simplifying the above expression, we get:
average = (12n + 96)/12
Simplifying further, we get:
average = n + 8
Therefore, the average of all twelve integers is n + 8.
Final Answer:
The average of all twelve integers is n + 8, which corresponds to option b) in the given choices.
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Consider a sequence of twelve consecutive even integers. If the average of the first seven is n, then the average of all the twelve integers is:a)nb)n + 5c)K xn, where K is a function of nd)n + (5/7)Correct answer is option 'B'. Can you explain this answer?
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