find the sum of the series: 1+2+3+4+5+7+8+9+11+13+15+16+17+...+64

To find the sum of the given series, we first need to observe the pattern of the numbers in the sequence. Let's analyze the series:

### Given Series:
\[ 1, 2, 3, 4, 5, 7, 8, 9, 11, 13, 15, 16, 17, \ldots, 64 \]

### Step 1: Identify the Pattern
The numbers in the series do not appear to follow a simple arithmetic or geometric progression. Instead, we notice that:

- The numbers in the series are all the natural numbers from 1 to 64, **except for multiples of 6** (6, 12, 18, ..., 60).

### Step 2: Find the Sum of All Numbers from 1 to 64
To find the sum of the series, we will:

1. Calculate the sum of all numbers from 1 to 64.
2. Subtract the sum of all multiples of 6 within the range from 1 to 64.

#### Sum of Numbers from 1 to 64
The sum of the first \(n\) natural numbers is given by the formula:
\[
S = \frac{n(n+1)}{2}
\]
For \(n = 64\):
\[
S = \frac{64 \times 65}{2} = 2080
\]

### Step 3: Find the Sum of Multiples of 6 up to 64
Now, we need to find the multiples of 6 up to 64:

Multiples of 6 in this range are: 6, 12, 18, 24, 30, 36, 42, 48, 54, 60.

This is an arithmetic series where the first term \(a = 6\), the common difference \(d = 6\), and the last term \(l = 60\).

The number of terms \(n\) in this arithmetic series is given by:
\[
n = \frac{l - a}{d} + 1 = \frac{60 - 6}{6} + 1 = 10
\]

The sum \(S_m\) of an arithmetic series is given by:
\[
S_m = \frac{n}{2} \times (a + l)
\]

Substitute the values:
\[
S_m = \frac{10}{2} \times (6 + 60) = 5 \times 66 = 330
\]

### Step 4: Find the Sum of the Given Series
Now, subtract the sum of the multiples of 6 from the sum of all numbers from 1 to 64:

\[
\text{Sum of the series} = 2080 - 330 = 1750
\]

### Conclusion
The sum of the series \(1 + 2 + 3 + 4 + 5 + 7 + 8 + 9 + 11 + 13 + 15 + 16 + 17 + \ldots + 64\) is **1750**.

Would you like more details, or do you have any questions?

Here are 5 related questions:
1. How do you find the sum of an arithmetic progression?
2. What are the properties of natural numbers?
3. How would the sum change if multiples of a different number were removed?
4. How can we calculate the sum of even numbers up to a certain limit?
5. What is the formula for the sum of squares of the first \(n\) natural numbers?

**Tip:** Understanding the patterns in sequences can help identify the rules governing them, which is essential for solving series-related problems.

Learn how to find the sum of a series from 1 to 64, excluding multiples of 6. This problem involves pattern recognition and arithmetic series formulas, ideal for students in Grades 6-8.

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