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.