Final answer:
The sum of the first 2023 natural numbers is 2046136, calculated using the formula for sum of natural numbers. The sum of the series ∑[k=1, 100] (1 / (k² + k)) simplifies to 100/101 after breaking it down into partial fractions and resulting in a telescoping series.
Step-by-step explanation:
To calculate ∑[k=1, 2023] k, we use the formula for the sum of the first n natural numbers:
S = n(n + 1) / 2, where S is the sum and n is the number of terms. Substituting 2023 for n, we get:
S = 2023(2023 + 1) / 2 = 2023 × 2024 / 2 = 2046136,
which gives us the sum from 1 to 2023.
For ∑[k=1, 100] (1 / (k² + k)), we can use partial fractions to break each term down:
1 / (k² + k) = 1 / k - 1 / (k + 1)
This results in a telescoping series. Summing this from k=1 to k=100, most terms will cancel out, and we are left with:
S = (1/1 - 1/2) + (1/2 - 1/3) + ... + (1/99 - 1/100) + (1/100 - 1/101)
The only terms that do not cancel are the first and the last, thus:
S = 1 - 1/101 = 100/101.