Question

I need help on this questions. These sequences are hard.​

Answer 1

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