Problem 44 - Algebra and Trigonometry with Analytic Geometry

Question

(a) Find the number of negative integers greater than \(-500\) that are divisible by 33 . (b) Find their sum.

Step-by-Step Solution

Answer

(a) 15 (b) -3960

Step 1: Define the Range of Integers

We are given that the integers are negative and greater than -500. This means we need to find integers in the range \(-499, -498, ext{...}, -1\)

Step 2: Determine Divisibility by 33

An integer is divisible by 33 if the integer can be expressed in the form \(33k\), where \(k\) is an integer. We need to find values of \(k\) such that \(33k < 0\) and \(33k > -500\).

Step 3: Find the First Negative Multiple of 33 Greater than -500

Check the multiples of 33 starting from -500 upwards.The first multiple we want is the smallest k such that \(33k > -500\). Compute: \(-500 \div 33 \approx -15.15\).Since \(k\) is an integer, we take \(k = -15\). The corresponding multiple is \(33(-15) = -495\).

Step 4: Identify All Negative Multiples of 33 Greater than -495

Continue adding 33 to -495 until you reach -1 or greater. This forms an arithmetic sequence: \(-495, -462, -429, -396, -363, -330, -297, -264, -231, -198, -165, -132, -99, -66, -33\).

Step 5: Count the Number of Integers

Count each element in this sequence: there are 15 numbers. Thus, there are 15 negative integers greater than -500 that are divisible by 33.

Step 6: Sum the Identified Integers

To find the sum of the sequence \(-495, -462, ..., -33\), use the formula for the sum of an arithmetic sequence: \[S_n = \frac{n}{2} (a_1 + a_n) \]where \(n = 15\), \(a_1 = -495\), and \(a_n = -33\).Compute: \[ S_{15} = \frac{15}{2} (-495 - 33) = \frac{15}{2} (-528) = -3960 \].

Key Concepts

Divisibility

Divisibility involves determining if one number can be divided by another without leaving a remainder. When dealing with this concept, a key approach is to express the number of interest as a product involving the divisor. For example, an integer is divisible by 33 if it can be expressed as \(33k\), where \(k\) is an integer.

• To verify divisibility, perform the division; if the remainder is zero, the number is divisible.


• This problem involves negative integers, so our goal is to find negative numbers that fit the divisibility condition.

To illustrate, consider finding negative multiples of 33 greater than -500. We compute \(-500 \div 33\) to approximate the boundary number, which reveals the first multiple \(-495\) when \(k = -15\). This step ensures you identify the starting point in your arithmetic sequence of divisible numbers.

Arithmetic Sequence

An arithmetic sequence is a series of numbers where each term increases by a constant difference known as the "common difference." This sequence can simplify finding specific terms or summing a series of terms. In our problem, the common difference is 33.
By identifying the initial term, -495, we know:

• The sequence starts at -495 and each subsequent number is produced by adding 33 to the previous one.


• Continuing this process generates terms up to -33 (i.e., -495, -462, -429, etc.).

This structure allows for easy counting, as each number in the constructed series is equally spaced. In this example, it provides the integers we seek, fully matched to our divisibility rule of 33, while remaining within the boundary conditions specified by the problem (greater than -500 and less than 0).

Sum of Integers

Calculating the sum of numbers in an arithmetic sequence can be done quickly using the formula \(S_n = \frac{n}{2} (a_1 + a_n)\), where \(S_n\) is the sum, \(n\) is the number of terms, \(a_1\) is the first term, and \(a_n\) is the last term.

• First, determine the count of terms (\(n\)) in the sequence. In this problem, there are 15 terms.


• Identify the first term (\(a_1 = -495\)) and the last term (\(a_n = -33\)).


• Plug these into the formula to compute the sum, which in this case is \(S_{15} = \frac{15}{2} (-495 - 33) = -3960\).

This method is efficient. Instead of adding each integer individually, the formula allows you to sum the sequence with minimal computation based on clear and established rules. Understanding and applying this formula is crucial for problems involving the sum of sequences.