Problem 16 - Basic Engineering Mathematics

Question

Find the sum of all the numbers between 0 and 207 which are exactly divisible by 3 .

Step-by-Step Solution

Answer

The sum is 7173.

Step 1: Identify the Divisible Numbers

First, identify the numbers between 0 and 207 that are divisible by 3. These numbers form an arithmetic sequence where the first term (a) is 3 and the common difference (d) is also 3.

Step 2: Calculate the Last Term

To find the last term (l) of this sequence that is under or equal to 207, divide 207 by 3 and round down to the nearest whole number. This gives 69, which means 3 multiplied by 69 is the last number divisible by 3. Hence, the last term is 207.

Step 3: Determine the Number of Terms

Use the formula for the number of terms in an arithmetic sequence: \( n = \frac{{l - a}}{d} + 1 \). Substitute \( a = 3 \), \( l = 207 \), \( d = 3 \). Therefore, \( n = \frac{{207 - 3}}{3} + 1 = 69 \).

Step 4: Calculate the Sum of the Sequence

The formula for the sum of an arithmetic sequence is \( S_n = \frac{n}{2} (a + l) \). Substitute \( n = 69 \), \( a = 3 \), \( l = 207 \). This gives \( S_{69} = \frac{69}{2} (3 + 207) = 7173 \).

Key Concepts

Divisibility

Divisibility is a fundamental concept in mathematics, particularly useful in problems involving arithmetic sequences. For a number to be divisible by another, it must leave a remainder of zero when divided. In the exercise, we focus on numbers divisible by 3. This means each number can be expressed as 3 multiplied by an integer.
Consider the example of finding numbers up to 207 divisible by 3. These numbers start from 3, 6, 9, and so on. Each of these numbers gives no remainder when divided by 3, making them part of the arithmetic sequence related to divisibility by 3. By understanding divisibility, you can easily identify members of such sequences, which simplifies many arithmetic calculations.

Sum of Sequence

The sum of a sequence, particularly an arithmetic sequence, is calculated using a specific formula. An arithmetic sequence has a common difference between consecutive terms. In this problem, the numbers we consider are divisible by 3 and they form the sequence: 3, 6, 9, ..., 207.
To find the sum of this sequence, we apply the formula:\[S_n = \frac{n}{2} (a + l)\]Where:

• \(S_n\) is the sum of the sequence,
• \(n\) is the number of terms,
• \(a\) is the first term, and
• \(l\) is the last term of the sequence.

By substituting into the formula, we find that the sum is 7173. Understanding how to compute this sum clarifies how arithmetic sequences work, especially when considering the designed range of divisible numbers.

Number of Terms

Determining the number of terms in an arithmetic sequence is straightforward with the right formula. The sequence in question is formed by numbers divisible by 3 up to 207. The formula to find the total number of terms is given by:\[n = \frac{{l - a}}{d} + 1\]Where:

• \(n\) is the number of terms,
• \(a\) is the first term,
• \(l\) is the last term, and
• \(d\) is the common difference.

Substituting the values gives:- \(a = 3\) (first term)- \(l = 207\) (last term)- \(d = 3\) (common difference)
This leads to 69 as the number of terms. Knowing this allows for quick tallying of terms in a sequence and is vital for accurately calculating sequence sums.