Compute the value of

\(1 - 2 + 3 - 4 + \dots + 2019 - 2020 + 2021 \)

Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator of the previous fraction by . He then multiplies all his fractions together. He has fractions, and their product equals . What is the value of the first fraction he wrote?

Find the value of the series

\(1 + 4 + 2 + 8 + 3 + 12 + 4 + 16 + \cdots + 24 + 96 + 25 + 100.\)

Guest May 14, 2022

#1**0 **

For the first one, you can look at the terms pairwise:

\((1 - 2) + (3 - 4) + (5 - 6) + \cdots + (2019 - 2020) + 2021\)

Each pair results in -1 and there are 1010 pairs of numbers (that is just 2020/2), so the answer is \((-1)(1010) + 2021 = 1011\).

Some numbers are missing from your second question. Please check again.

For the third question, look at the terms pairwise again:

\(\quad(1 + 4) +(2 + 8) + (3 + 12) + (4 + 16) + \cdots + (24 + 96) + (25 + 100) \\ =5 + 10 + 15 + 20 + \cdots + 120 + 125\)

Now that is just the sum of an arithmetic sequence. You can find the first term, common difference, and the number of terms, then plug it into the formula to calculate it.

MaxWong May 14, 2022

#2**0 **

Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 1 greater than the denominator. He then writes several more fractions. To make each new fraction, he increases both the numerator and the denominator of the previous fraction by 1. He then multiplies all his fractions together. He has 20 fractions, and their product equals 3. What is the value of the first fraction he wrote?

Guest May 14, 2022