Introduction to Summation of Specific Number Sequences

Finding the sum of specific number sequences, such as odd numbers, can be an interesting exercise in mathematical pattern recognition and formula application. In this article, we will explore how to calculate the sum of three distinct sequences of numbers. These sequences are: 3-digit odd numbers starting with a non-zero digit, odd numbers less than 1000, and numbers less than 100 ending in 5 or 7. We will break down each sequence and provide the step-by-step calculations for their respective sums.

Sum of 3-Digit Odd Numbers with No Zeros in Front

First, we tackle the problem of finding the sum of 3-digit odd numbers where the first digit is not zero.

To find the sum, we follow these steps:

Calculate the total sum of all 1-digit, 2-digit, and 3-digit odd numbers. Subtract the sum of all 1-digit and 2-digit odd numbers from the total sum calculated in step 1.

The formula for the sum of the first n odd numbers is (n^2). Since we are dealing with odd numbers, we can use this formula directly.

Step 1: Calculate the total sum of all 1-digit, 2-digit, and 3-digit odd numbers.

[frac{1000}{2} 500] [text{Sum} 500^2 250,000]

Step 2: Calculate the sum of all 1-digit and 2-digit odd numbers.

[frac{100}{2} 50] [text{Sum} 50^2 2,500]

Subtract the sum of 1-digit and 2-digit numbers from the sum of 1-digit, 2-digit, and 3-digit odd numbers:

[text{Sum of 3-digit odd numbers} 250,000 - 2,500 247,500] Note: The formula works because it eliminates the even numbers from the total sum, leaving us with only the odd numbers.

Sum of Odd Numbers Less Than 1000

Next, we calculate the sum of all odd numbers less than 1000.

The process is similar to the previous calculation:

Calculate the total sum of all 1-digit, 2-digit, and 3-digit odd numbers. This time, the total sum is the same as before because we are including both 1-digit and 2-digit numbers.

Using the same formula:

[frac{1000}{2} 500] [text{Sum} 500^2 250,000]

Sum of Numbers Less Than 100 Ending in 5 or 7

Finally, we find the sum of numbers less than 100 that end in 5 or 7.

This problem requires breaking down the sequence and summing the unit and tens digits separately:

Count the number of numbers ending in 5 and 7. Sum the unit digits of these numbers. Sum the tens digits of these numbers.

Step 1: Count the numbers ending in 5 and 7.

For numbers ending in 5: [10 times 5 50] (one on each set of 10 numbers from 1 to 99) For numbers ending in 7: [10 times 7 70] (one on each set of 10 numbers from 1 to 99)

Step 2: Sum the unit digits.

[50 70 120]

Step 3: Sum the tens digits.

For each set of 10 numbers, there are 9 tens digits, and we need to account for 9 tens digits for both 5 and 7:

[text{Sum of tens digits} (1 2 3 4 5 6 7 8 9) times 2 times 10] [text{Sum of tens digits} 45 times 2 times 10 900]

Add the unit digit sum and the tens digit sum:

[120 900 1020]

Conclusion

In summary, the sum of 3-digit odd numbers with no zeros in front is 247,500, the sum of odd numbers less than 1000 is 250,000, and the sum of numbers less than 100 ending in 5 or 7 is 1020. Understanding the underlying patterns and applying the appropriate formulas can help simplify these calculations significantly.

Key Takeaways:

The sum of 3-digit odd numbers with no zeros in front: 247,500 The sum of odd numbers less than 1000: 250,000 The sum of numbers less than 100 ending in 5 or 7: 1020