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
4. Giraffe's Neck: Lamarck's Theory
A giraffe cannot pass on a longer neck just because it stretches it to reach leaves. Acquired traits are not inherited. Only genetic changes passed from parents to offspring can be inherited.
5. South American Rhea and African Ostrich: Natural Selection
The populations became separated and lived in different environments. Over many generations, different inherited traits helped them survive and reproduce. Eventually, the populations became two different species.
6. Comparing Lamarck's and Darwin's Theories
Comparison | Lamarck's Theory | Darwin's Theory |
|---|---|---|
Origin of new traits | Traits develop through use and disuse. | Helpful inherited variations become more common through natural selection. |
Inheritance of traits | Acquired traits are passed to offspring. | Genetic traits are passed from parents to offspring. |
Species extinction | Species change when their needs and environments change. | Species may become extinct if they cannot survive environmental changes. |
Main driving force | Use and disuse of body parts. | Natural selection and competition for survival. |
7. Which Individuals Are at a Disadvantage, and Why?
Birds with medium-sized beaks may be at a disadvantage because medium-sized seeds are scarce. Birds with small beaks can eat soft seeds, while birds with large beaks can crack hard seeds.
8. Type of Natural Selection
Disruptive selection may occur because birds with small or large beaks can have an advantage when medium-sized seeds are scarce.
9. Environmental Change and Natural Selection
The scarcity of medium-sized seeds may favor birds with small or large beaks. Over many generations, these traits may become more common because those birds can find food more easily.
Important: Answers 8 and 9 are provisional, because the graph does not clearly confirm disruptive selection. The other answers are suitable for the workshop and use simple vocabulary.