There are 9 consecutive integers in a certain sequence. If the average

Question

There are 9 consecutive integers in a certain sequence. If the average of the first seven integers in the sequence is n, what is the average of all 9 integers in the sequence?

A) n
B) n-1
C) n+1
D) 2n
E) n(n+1)

BrentGMATPrepNow · Accepted Answer

USEFUL RULE: In a set of EQUALLY SPACED numbers, the mean of the set equals the median of the set

Let O, O, O, O, O, O, O, O, O represent the 9 consecutive integers listed in  ascending order .

The average (aka MEAN) of the first seven integers in the sequence is n  
Here are the first seven integers: O, O, O, O, O, O, O
Since consecutive integers are equally spaced, the mean = the median
The median of these 7 numbers is the middlemost number. 
So, the middlemost value is also the mean. 
We get: O, O, O, n, O, O, O
So, our NINE numbers now look like this: O, O, O, n, O, O, O, O, O

What is the average of all 9 integers in the sequence?  
Using the same logic, we can see that the mean of the 9 numbers is the middlemost number. 
So, replace the middlemost number with a " ? " : O, O, O, n,  ? , O, O, O, O

We can see that  ?  (the mean of all 9 integers) is  1 GREATER THAN  n (the mean of the first 7 integers)
In other words, the mean of all 9 integers = n+1

Answer: C

pushpitkc · Answer

Consider the 9 consecutive number 1,2,3,4,5,6,7,8,9

The average of the first seven integers is 4(Middle term, since elements are in AP)
The average of the entire lot will be 5(which is n+1, provided the average of first 7 numbers is n)

Hence,  Option C  is the answer option.

Abhishek009 · Answer

Plug in and check...
 
Let the sequence of consecutive integers be   1 , 2 , 3 , 4 , 5 , 6 , 7  ,  8 , 9

Sum of first 7 numbers is 28 
Average of first 7 numbers is  \frac{28}{7} = 4

Sum of first 9 numbers is 45
Average of first 9 numbers is  \frac{45}{9} = 5

Hence, the answer will be (C) n+1

gracie · Answer

if a=average of all 9 consecutive integers, 
a will be the 5th term
n will be the 4th term
a=n+(5-4)=n+1
C

pcs140 · Answer

Since we know n is average for 7, for 9 consecutive numbers it would be n+ (n+1) +(n+2) / 3 = 3n+3/3. which is n+1.
Ans C.

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There are 9 consecutive integers in a certain sequence. If the average

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There are 9 consecutive integers in a certain sequence. If the average

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