If the average (arithmetic mean) of seven consecutive integers is k +

Let first integer of series = a
Difference between integers = d= 1 (consecutive integers given)
Here no. of terms in series = n=7
Average = K+2


Average = Arithmetic mean =\(\frac{sum of all 7 numbers}{7}\)

by formula Sum of series =\(\frac{n(2a+(n-1)d)}{2} = \frac{7(2a+(7-1)*1)}{2} = \frac{7(2a+6)}{2} = 7(a+3)\)

As Average = k+2 = \(\frac{sum of all 7 numbers}{7}\) = \(\frac{7(a+3)}{7}\)
=> k+2 =a+3
=> a= k-1

Now as nth term of series = a+(n-1)d => last term of given series = a+(7-1)*1 = (k-1)+6 = k+5

Product of first and last term = (k-1)*(k+5) = k^2+5k-k-5 = k^2+4k-5

Answer: C

Given data: A set with 7 consecutive numbers has mean k+2

Lets assume the 7 numbers to be 1,2,3,4,5,6,7
The mean of consecutive integers in a set with odd number of elements is the middle number.
In this set, the 4th element which is 4 is the mean.
k+2 = 4 => k = 2
The product of the least and the greatest number is 1*7(7)

Evaluating the answer options(s.t k = 2)
A. k^2 - 9 = 4 -9 = -5
B. k^2 - 2k + 1 = 4 - 4 + 1 = 1
C. k^2 + 4k - 5 = 4 + 8 - 5 = 7

Since Option C gives us the same value for the product of the smallest and greatest number,
we need not evaluate the other 2 answer options. (Option C) is our answer!

Total 7 consecutive nos and K+2 is the average which means 3 nos each are on the right and left of this no.
K-1, k,k+1,k+2,k+3, k+4, k+5
Smallest no=k-1
largest no=k+5
Product=(k-1)(k+5)=k²+4k-5=C is the answer

I used algebra, erred, and took a long time to correct. So I timed myself using what I thought was a time-consuming method. Wrong. I was well under a minute.

I used only one sketch, and no fancy lines or asterisks. I used them here for clarity.

1. List the integers

x|x+1|x+2|x+3|x+4|x+5|x+6|

2. Place (k+2) under the median

"The arithmetic mean is k + 2." Median = mean for evenly spaced set. So

x|x+1|x+2|x+3|x+4|x+5|x+6|
*|***|***|k+2|***|***|***|

3. Find k. From above:
k + 2 = x + 3
k = x + 1, so

x|x+1|x+2|x+3|x+4|x+5|x+6|
*|_k_|***|k+2|***|***|***|

4. Find answer. The product of the greatest and least integer is? We need x, and (x+6), in terms of k, so

_x_|x+1|x+2|x+3|x+4|x+5|x+6|
k-1|_k_ |***|k+2|***|***|k+5|

(k - 1)(k + 5) = k\(^2\) + 4k - 5

ANSWER C

Is this a 500 lvl question? really doesn't feel like it

anyway

Sum of the numbers = 7k + 14
let x = smallest number
21 + x = 7k + 14
x= k-1

product of smallest and biggest = x(x+6) = x^2 + 6x = k^2 - 2k + 1 + 6k - 6 = k^2 +4k - 5

Thus, answer C

Not a hard problem, but I don't think it is level 500.

the source is from Magoosh

Added the source. Thank you.

Official Explanation:

For any collection of evenly spaced numbers, the mean and median are always equal. Because this is set with seven members, an odd number of elements, the median is the middle number, the 4th number on the list: there are three numbers on the list below it and three above it. Thus, (k + 2) is the mean as well as the median, the middle or 4th number on the list. Since there are only seven numbers, we can simply write them out:

{k – 1, k, k + 1, k + 2, k + 3, k + 4, k + 5}

The least, three lower than the median, is (k – 1).

The greatest, three higher than the median, is (k + 5).

Their product is:

product = (k – 1)(k + 5) = k2 + 4k – 5

Answer = (E)

in OE they have assumed 7 numbers as {k – 1, k, k + 1, k + 2, k + 3, k + 4, k + 5} However I think in such cases right assumption should be {k – 3, k - 2 , k - 1, k, k + 1, k + 2, k + 3} this will simplify the sum to 7K. In complex situation this will be quiet helpful.

We can let the smallest integer in the set = x and the largest = x + 6; thus:

(x + x + 6)/2 = k + 2

2x + 6 = 2k + 4

2x = 2k - 2

x = k - 1

Thus, the smallest integer is k - 1 and the greatest is k + 5, so the product of those two values is:

(k - 1)(k + 5) = k^2 + 4k - 5

Alternate solution:

Since k + 2 is in the average of the 7 consecutive integers, it’s also the median, i.e., the 4th integer. Thus, the largest integer is 3 more than k + 2, i.e., k + 5, and the least integer is 3 less than k + 2, i.e., k - 1. The product of the greatest and least integer is:

(k + 5)(k - 1) = k^2 + 4k - 5

Answer: C

could you explain how the greatest integer = k+5 and the next step?


I am confused

Theory: In case of evenly spaced sets the mean of the set is the middle number (in case of odd number of numbers)

The average (arithmetic mean) of seven consecutive integers is k + 2
=> Middle number (or the \(4^{th}\) number) = k+2

So, the consecutive numbers less than k+2 are
k-1, k, k+1

The consecutive numbers greater than k+2 are
k+3, k+4, k+5

The product of the greatest and least integer
= (k-1) *(k+5)
= \(k^2\) - k + 5k - 5
= \(k^2\) + 4k - 5

So, Answer will be C
Hope it helps!

To learn more about Statistics watch the following video

Initially, ignore K, and find out the arithmetic mean keeping in mind that the integers are consecutive:

(-1) + (0) + (1) + (2) + (3) + (4) + (5)/7= 2

Now, just add K with the greatest and the least number to find out the their product: (K-1) (K+5) = K[*2]+4K-5

Therefore, C is the correct answer.