a) Levans writes a positive fraction in which the numerator and denominator are integers, and the numerator is 2 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 10 fractions, and their product equals 7/2. What is the value of the first fraction he wrote?
b) Compute \(1 + \frac{3}{5} + \frac{5}{5^2} + \frac{7}{5^3} + \dotsb.\)
c) Compute \(\frac{1}{1 \times 4} + \frac{1}{4 \times 7} + \frac{1}{7 \times 10} + \dots + \frac{1}{97 \times 100}\)
d) We derived that \(\frac{1}{n(n + 1)} = \frac{1}{n} - \frac{1}{n + 1}.\)
Fill in the blanks to make a true equation: \(\frac{1}{n(n+3)}= \frac{blank}{n}+\frac{blank}{n+3}\)
Please include an explanation for each of these, thanks!
a)
The first fraction is 3/1.
The product of the 10 fractions is 7/2. This means that the numerator of the product is 7 and the denominator is 2. The numerator of the first fraction is 2 greater than the denominator, so the numerator is 3 and the denominator is 1.
See a detailed answer to b) here: https://web2.0calc.com/questions/even-more-more-more-geometric-series-and-sequences
Here is a fairly detailed answer to c):
a=listfor(n, 0, 32 ,1 / ((1+3*n)*(1+3*(n+1)));print a, "==",sum a
(1 / 4, 1 / 28, 1 / 70, 1 / 130, 1 / 208, 1 / 304, 1 / 418, 1 / 550, 1 / 700, 1 / 868, 1 / 1054, 1 / 1258, 1 / 1480, 1 / 1720, 1 / 1978, 1 / 2254, 1 / 2548, 1 / 2860, 1 / 3190, 1 / 3538, 1 / 3904, 1 / 4288, 1 / 4690, 1 / 5110, 1 / 5548, 1 / 6004, 1 / 6478, 1 / 6970, 1 / 7480, 1 / 8008, 1 / 8554, 1 / 9118, 1 / 9700) == 33 / 100
