Step-by-step explanation:
1. "List the first 9 terms of the sequence defined recursively by
Sₙ = Sₙ₋₂ · (Sₙ₋₁ − 1)
with S₁ = 2 and S₂ = 3"
The first two terms are provided. To find the third term, substitute n = 3:
S₃ = S₃₋₂ · (S₃₋₁ − 1)
S₃ = S₁ · (S₂ − 1)
S₃ = 2 · (3 − 1)
S₃ = 4
Repeating this process, we get that the first nine terms of the sequence are:
2, 3, 4, 9, 32, 279, 8896, 2481705, 22077238784
2. "Compute the value of ∑(k=1 to 8) Sk, where Sk is the same sequence as in the previous question."
All we have to do here is add up the first 8 terms:
2 + 3 + 4 + 9 + 32 + 279 + 8896 + 2481705
= 2490930
3. "Is the sequence from question 1 arithmetic, geometric, or neither? Explain how you reach your conclusion."
If the sequence were arithmetic, it would have a common difference. But the difference between 3 and 4 is not the same difference between 4 and 9.
If the sequence were geometric, it would have a common ratio. But the ratio between 2 and 3 is not the same ratio between 3 and 4.
So the series is neither arithmetic nor geometric.
4. "Find the sum of the first 200 terms of the arithmetic sequence that begins: 12, 18, 24..."
The sum of the first n terms of an arithmetic sequence is:
S = n (a₁ + aₙ) / 2
where a₁ is the first term and aₙ is the nth term.
The nth term of an arithmetic sequence is:
aₙ = a₁ + d (n − 1)
where d is the common difference.
Here, the first term is 12, so a₁ = 12. The common difference is 18 − 12 = 6. So the 200th term is:
a₂₀₀ = 12 + 6 (200 − 1)
a₂₀₀ = 1206
The sum of the first 200 terms is:
S = 200 (12 + 1206) / 2
S = 121800
