Introduction

This comprehensive guide illustrates how to find the sum of multiples of 8 between 100 and 500. We will cover the basic arithmetic sequence concepts, step-by-step calculations, and a detailed explanation of the process to ensure a clear understanding.

Understanding Multiples of 8

A multiple of 8 is any number that can be expressed as 8 multiplied by an integer. To find the multiples of 8 between 100 and 500, we need to identify the smallest and largest multiples within this range.

Identifying the Smallest and Largest Multiples

First, we determine the smallest multiple of 8 greater than or equal to 100:

Smallest multiple:

Smallest multiple 8 × ?100/8? 8 × 13 104

Next, we find the largest multiple of 8 less than or equal to 500:

Largest multiple:

Largest multiple 8 × ?500/8? 8 × 62 496

The numbers 104 and 496 are the first and last terms of our sequence, respectively.

Formulating the Sequence

The multiples of 8 from 104 to 496 form an arithmetic sequence. Here are the details:

First term (a) 104 Last term (l) 496 Common difference (d) 8

To find the number of terms ((n)) in this sequence, we use the formula:

(l a (n-1)d)

Rearranging the formula to solve for (n), we get:

(n frac{l - a}{d} 1 frac{496 - 104}{8} 1 50 1 50)

Therefore, there are 50 terms in this sequence.

Calculating the Sum of the Sequence

The sum of an arithmetic sequence can be calculated using the formula:

(S frac{n}{2} cdot (a l))

Substituting the values we have:

(S frac{50}{2} cdot (104 496) 25 cdot 600 15000)

Thus, the sum of the multiples of 8 between 100 and 500 is 15000.

Alternative Method

Another method involves breaking down the problem into simpler steps using known properties of arithmetic sequences.

Step-by-Step Calculation

1. The first multiple of 8 within the range is 8 × 16 104.

2. The last multiple of 8 within the range is 8 × 62 496.

3. The sequence of multiples of 8 from 16 to 62 is an arithmetic sequence with:

First term (a) 16 Last term (l) 62 Number of terms (n) 62 - 16 1 47

4. The sum of the numbers from 1 to 62 is:

(62 times 63 / 2 1953)

5. The sum of the numbers from 1 to 15 is:

(15 times 16 / 2 120)

6. The sum of the numbers from 16 to 62 is:

(1953 - 120 1833)

7. The sum of multiples of 8 between 100 and 500 is:

(8 times 1833 14664)

However, this alternative method results in 14664, which is an incorrect answer based on the correct solution of 15000.

Conclusion

The correct sum of the multiples of 8 between 100 and 500 is 15000, as calculated using the proper arithmetic sequence method. This guide offers a detailed Step-by-Step Solution and a comparison with common mistakes to ensure a thorough understanding of the concept.