Set T consists of 82 consecutive odd integers. If the sum of the integers is

[BTGModeratorVI]

Set T consists of 82 consecutive odd integers. If the sum of the integers is 3^8 - 165, what is the median of set T?

A) 78
B) 79
C) 80
D) 81
E) 82

Answer: A
Source: GMAT Prep Now

[Brent@GMATPrepNow]

Set T consists of 82 consecutive odd integers. If the sum of the integers is 3^8 - 165, what is the median of set T?

A) 78
B) 79
C) 80
D) 81
E) 82

Answer: A
Source: GMAT Prep Now

There's a nice rule that says, "In a set where the numbers are equally spaced, the mean will equal the median."
For example, in each of the following sets, the mean and median are equal:
{7, 9, 11, 13, 15}
{-1, 4, 9, 14}
{3, 4, 5, 6}

Since set T consists of 82 consecutive odd integers (e.g., 5, 7, 9, 11, etc), we can see that these values are equally spaced.
So, the mean and median of set T are equal.

So, to find the answer, we need only find the mean of set T.

Median = mean = (3⁸ - 165)/82
= (3⁸ - 1 - 164)/82 [you'll see why I did this shortly]
= (3⁸ - 1)/82 - 164/82
= (3⁸ - 1)/82 - 2 [PERFECT!! 164/82 simplified to 2]

Now let's do some factoring [since 3⁸ - 1 is a DIFFERENCE OF SQUARES]
= (3⁴ + 1)(3⁴ - 1)/82 - 2
= (81 + 1)(81 - 1)/82 - 2
= (82)(80)/82 - 2
= 80 - 2
= 78

Answer: A

Cheers,
Brent

[Scott@TargetTestPrep]

Set T consists of 82 consecutive odd integers. If the sum of the integers is 3^8 - 165, what is the median of set T?

A) 78
B) 79
C) 80
D) 81
E) 82

Answer: A

Solution:

In a set of consecutive odd integers, which are evenly spaced, the median of the set is also the average of the set. Therefore, if we let x be the median, we can create the equation:

82x = 3^8 - 165

82x = 3^8 - 1 - 164

82x = (3^4 - 1)(3^4 + 1) - 164

82x = (80)(82) - 2(82)

x = 80 - 2

x = 78

Answer: A