Enter a count to quickly calculate the sum, sum of squares, and sum of cubes for the first few natural numbers, even numbers, odd numbers, prime numbers, or composite numbers.
Natural numbers are the set of positive integers. The sum of the first \( n \) natural numbers can be calculated using the formula: \( S = \frac{n(n+1)}{2} \)
This formula is derived from the arithmetic sequence, where natural numbers form an arithmetic sequence with a common difference of 1. For example, to find the sum of the first 5 natural numbers: \( S = 1 + 2 + 3 + 4 + 5 = 15 \) Using the formula to verify: \( S = \frac{5(5+1)}{2} = \frac{5 \times 6}{2} = 15 \)
Even numbers are divisible by 2. The sum of the first \( n \) even numbers is calculated using the formula: \( S = n(n+1) \) For example, to find the sum of the first 5 even numbers: \( S = 2 + 4 + 6 + 8 + 10 = 30 \) Using the formula to verify: \( S = 5(5+1) = 5 \times 6 = 30 \)
Odd numbers are not divisible by 2. The sum of the first \( n \) odd numbers is calculated using the formula: \( S = n^2 \) For example, to find the sum of the first 5 odd numbers: \( S = 1 + 3 + 5 + 7 + 9 = 25 \) Using the formula to verify: \( S = 5^2 = 25 \)
Prime numbers are natural numbers greater than 1 that have no divisors other than 1 and themselves. There is no simple formula to sum prime numbers, so each prime number must be added individually. For example, the first 5 prime numbers are 2, 3, 5, 7, and 11, and their sum is: \( S = 2 + 3 + 5 + 7 + 11 = 28 \)
Composite numbers are natural numbers with more than two divisors. The first 5 composite numbers are 4, 6, 8, 9, and 10, and their sum is: \( S = 4 + 6 + 8 + 9 + 10 = 37 \)
Solution:
\( S = \frac{10(10+1)}{2} = \frac{10 \times 11}{2} = 55 \)
Result: Sum of the first 10 natural numbers = 55
Solution:
\( S = 6(6+1) = 6 \times 7 = 42 \)
Result: Sum of the first 6 even numbers = 42
Solution:
\( S = 7^2 = 49 \)
Result: Sum of the first 7 odd numbers = 49
Solution:
\( S = 2 + 3 + 5 + 7 + 11 + 13 + 17 + 19 = 77 \)
Result: Sum of the first 8 prime numbers = 77
Solution:
\( S = 4 + 6 + 8 + 9 + 10 + 12 + 14 + 15 + 16 + 18 = 112 \)
Result: Sum of the first 10 composite numbers = 112