Find the value of the sum 3+5+6+9+10+12+15+18+20.............+100

To find the sum \( 3 + 5 + 6 + 9 + 10 + 12 + 15 + 18 + 20 + \ldots + 100 \), let's analyze the sequence.

### Step 1: Identify the sequence pattern
- The sequence starts with \( 3 \).
- The difference between consecutive terms doesn't follow a simple arithmetic pattern.
- However, we can observe that the numbers are either multiples of 3 or 5.

Thus, the sequence consists of numbers that are multiples of either 3 or 5, starting from 3 and ending at 100.

### Step 2: Sum the multiples of 3
Multiples of 3 from 3 to 100: 
\[ 3, 6, 9, \ldots, 99 \]

This is an arithmetic series with the first term \( a = 3 \), the common difference \( d = 3 \), and the last term \( l = 99 \).

The number of terms \( n \) in this sequence is given by:
\[
n = \frac{l - a}{d} + 1 = \frac{99 - 3}{3} + 1 = 33
\]

The sum \( S_3 \) of the multiples of 3 is:
\[
S_3 = \frac{n}{2} \times (a + l) = \frac{33}{2} \times (3 + 99) = 33 \times 51 = 1683
\]

### Step 3: Sum the multiples of 5
Multiples of 5 from 5 to 100:
\[ 5, 10, 15, \ldots, 100 \]

This is also an arithmetic series with the first term \( a = 5 \), the common difference \( d = 5 \), and the last term \( l = 100 \).

The number of terms \( n \) in this sequence is:
\[
n = \frac{l - a}{d} + 1 = \frac{100 - 5}{5} + 1 = 20
\]

The sum \( S_5 \) of the multiples of 5 is:
\[
S_5 = \frac{n}{2} \times (a + l) = \frac{20}{2} \times (5 + 100) = 10 \times 105 = 1050
\]

### Step 4: Subtract the sum of multiples of 15
Multiples of 15 (which are multiples of both 3 and 5) from 15 to 90:
\[ 15, 30, 45, \ldots, 90 \]

This is an arithmetic series with the first term \( a = 15 \), the common difference \( d = 15 \), and the last term \( l = 90 \).

The number of terms \( n \) in this sequence is:
\[
n = \frac{l - a}{d} + 1 = \frac{90 - 15}{15} + 1 = 6
\]

The sum \( S_{15} \) of the multiples of 15 is:
\[
S_{15} = \frac{n}{2} \times (a + l) = \frac{6}{2} \times (15 + 90) = 3 \times 105 = 315
\]

### Step 5: Apply the inclusion-exclusion principle
The sum of all terms in the original sequence is given by:
\[
S = S_3 + S_5 - S_{15} = 1683 + 1050 - 315 = 2418
\]

So, the value of the sum is \( \boxed{2418} \).

Would you like further details or explanations on any of the steps? Here are 8 related questions you might consider:

1. How do we find the sum of an arithmetic series?
2. What is the inclusion-exclusion principle?
3. How do you determine the number of terms in an arithmetic sequence?
4. Can you explain how multiples of two different numbers interact in a sequence?
5. How would you calculate the sum if the sequence only had multiples of 3 or 5?
6. What if the sequence went beyond 100? How would the approach change?
7. Could we use a different method, like direct summation, for this problem?
8. How does this method compare to finding the sum of multiples in other ranges or step sizes?

**Tip:** Remember, for sequences involving multiples of different numbers, always account for overlaps using the inclusion-exclusion principle to avoid counting duplicates.

Discover how to find the sum of multiples of 3 and 5 up to 100 using the arithmetic series sum formula and the inclusion-exclusion principle. This problem involves identifying multiples of 3 and 5, calculating their sums separately, and subtracting the overlap to get the final result. Suitable for advanced high school students and those interested in arithmetic series and mathematical problem-solving.

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