Maggie and Jayden go hiking once each month. During the first month they hike 0.5 miles, and then they double the length of their hike each subsequent month. Use this geometric series to manually sum how many miles they walk after the first six months: E^6 n=1 (0.5)(2)^n-1.
To manually sum this geometric series, we can use the formula for the sum of the first n terms of a geometric series:
S_n = a(1 - r^n) / (1 - r)
In this case, a = 0.5 (the first term of the series) and r = 2 (the common ratio).
So for n = 6:
S_6 = 0.5(1 - 2^6) / (1 - 2)
S_6 = 0.5(1 - 64) / -1
S_6 = 0.5(-63) / -1
S_6 = -31.5 / -1
S_6 = 31.5
Therefore, after the first six months, Maggie and Jayden would have hiked a total of 31.5 miles.