Sum of Numbers Divisible by 3 Between 1 and 34: A Comprehensive Guide

When you need to find the sum of numbers divisible by 3 between 1 and 34, it's a classic example that involves understanding arithmetic series. This guide will walk you through the process, explaining the concepts step-by-step.

Understanding the Problem

The problem states: Find the sum of the numbers between 1 and 34 that are divisible by 3. The numbers in this range that fulfill the criterion are 3, 6, 9, 12, 15, 18, 21, 24, 27, 30, and 33. These numbers form an arithmetic sequence.

Identify the Numbers Divisible by 3

First, we identify the numbers in the sequence:

3, 6, 9, 12, 15, 18, 21, 24, 27, 30, 33

The smallest number is 3, and the largest is 33.

Count the Terms

To find the number of terms in this sequence, we can use the formula for the n-th term of an arithmetic sequence:

l a (n - 1)d

Here, l is the last term, a is the first term, d is the common difference, and n is the number of terms. Plugging in the values:

33 3 (n - 1)3

30 3n - 3

33 3n - 3

36 3n

n 12

After solving, we find that n 11, which means there are 11 terms in the sequence.

Calculate the Sum of the Series

The sum of the first n terms of an arithmetic series can be calculated using the formula:

Sn n/2 (a l)

Substituting n 11, a 3, and l 33:

S11 11/2 (3 33) 11/2 (36) 11 * 18 198

This confirms that the sum of the numbers between 1 and 34 that are divisible by 3 is 198.

Alternative Method

Another approach involves pairing the numbers to simplify the calculation. Here's a step-by-step guide:

Identify the first number and the last number: 3 (first) and 33 (last). Notice that there are 11 numbers in this set. Divide the number of terms by 2 to get 5.5. Add the first number and the last number: 3 33 36. Multiply by the number of high-low pairs: 36 * 5.5 198.

This method also arrives at the same result, making the calculation straightforward and efficient.

Conclusion

Both methods confirm that the sum of numbers between 1 and 34 that are divisible by 3 is 198. This problem helps reinforce the concepts of arithmetic series and provides a practical illustration of how to apply the sum formula.

Related Topics

1. Arithmetic Series: An introduction to arithmetic sequences and series, including formulas and examples.

2. Sum of Numbers: Methods and techniques for finding the sum of a series of numbers.

3. Divisibility: Understanding divisibility rules and their applications in number theory.