Sum of All Numbers Between 500 and 900 Not Divisible by 7

Understanding and calculating the sum of specific numbers within a range, especially those that meet certain criteria, is a valuable skill in mathematics. This article will guide you through finding the sum of all numbers between 500 and 900 that are not divisible by 7, which involves several steps and formulas commonly used in arithmetic series calculations.

Arithmetic Series and Sum Formula

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

(S_n frac{n}{2} times (a l))

n - number of terms a - first term l - last term

Step-by-Step Calculation

Step 1: Sum of All Numbers from 500 to 900

a 500 l 900

To find the number of terms:

(n frac{l - a}{1} 1 900 - 500 1 401)

Now, calculating the sum:

(S frac{401}{2} times (500 900) frac{401}{2} times 1400 401 times 700 280,700)

Step 2: Sum of Numbers Divisible by 7 from 500 to 900

First multiple of 7 greater than or equal to 500: 504 (since (500 div 7) 71.43, and (71 times 7 497)) Last multiple of 7 less than or equal to 900: 896 (since (900 div 7) 128.57, and (128 times 7 896))

The sequence is: 504, 511, 518, ..., 896. This is an arithmetic series with:

a 504 d 7 l 896

Using the formula to find the number of terms:

(896 504 (n-1) times 7) ((896 - 504) (n-1) times 7) ((392) (n-1) times 7) (n-1 56) (n 57)

Now, calculating the sum of multiples of 7:

(S frac{57}{2} times (504 896) frac{57}{2} times 1400 57 times 700 39,900)

Step 3: Calculate the Sum of Numbers Not Divisible by 7

The final calculation is to subtract the sum of the multiples of 7 from the total sum:

(280,700 - 39,900 240,800)

Conclusion

Thus, the sum of all numbers between 500 and 900 that are not divisible by 7 is 240,800. This method can be applied to other similar problems, providing a structured and efficient approach to solving such arithmetic series challenges.

Related Keywords

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