Set S consists of n consecutive integers. The sum of the smallest three integers in set S … — GMAT

Set $S$ consists of $n$ consecutive integers. The sum of the smallest three integers in set $S$ is $-33$ , and the sum of the largest three integers in set $S$ is $69$ . What is the value of $n$ ?

Answer: 37

Answer

The total number of consecutive integers in set

S

is 37.

Solving

3x + 3 = -33

yields a first term of

-12

, and solving

3y - 3 = 69

yields a last term of

24

. The total count of consecutive integers in an inclusive range is

\text{last} - \text{first} + 1

, giving

24 - (-12) + 1 = 37

Step-by-Step Solution

Find the smallest integer in set S

The smallest integer is -12

Let the smallest integer be

x

. The sum of the smallest three consecutive integers is

x + (x + 1) + (x + 2) = 3x + 3

. Setting

3x + 3 = -33

yields

3x = -36

, so

x = -12

Find the largest integer in set S

The largest integer is 24

Let the largest integer be

y

. The sum of the largest three consecutive integers is

(y - 2) + (y - 1) + y = 3y - 3

. Setting

3y - 3 = 69

yields

3y = 72

, so

y = 24

Calculate the total number of elements n in set S

n = 37

For an inclusive set of consecutive integers bounded by first term

a

and last term

b

, the total number of terms is

n = b - a + 1

. Here,

n = 24 - (-12) + 1 = 37

Key Concept

Counting inclusive terms in a sequence of consecutive integers

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