Math Solutions and Lamarck vs. Darwin Evolution Answers
1. Arithmetic Sequence Calculations
A) d = u2 − u1
B) un = u1 + (n − 1)d
C) Sn = n/2 [2u1 + (n − 1)d]
d = 52 − 55 = −3
u10 = 55 + (10 − 1)(−3)
u10 = 55 + 9(−3)
u10 = 55 − 27
u10 = 28
S8 = 8/2 [2(55) + (8 − 1)(−3)]
S8 = 4 [110 + 7(−3)]
S8 = 4(110 − 21)
S8 = 4(89)
S8 = 356
2. Matchstick Pattern
un = 4 + (n − 1)3, so un = 3n + 1
A) Figure 5: u5 = 3(5) + 1 = 15 + 1 = 16 matchsticks
Figure 6: u6 = 3(6) + 1 = 18 + 1 = 19 matchsticks
B) u10 = 3(10) + 1 = 30 + 1 = 31 matchsticks
C) 3n + 1 = 200
3n = 200 − 1
3n = 199
n = 199/3 ≈ 66.33
Figure 66 requires 199 matchsticks, which is the closest value to 200.
3. Arithmetic Sequence with u1 = 1, un = 61, d = 2
un = u1 + (n − 1)d
61 = 1 + (n − 1)2
61 − 1 = 2(n − 1)
30 = n − 1
n = 31 terms