Question

1. We define a sequence of real numbers recursively by s_{1}=1, s_{2}=1 . and for 3, s_{n}=2 s_{n-1}-2 s_{n-2} (a) Write out the first 12 terms of this sequence.

Answer 1

To find the first 12 terms of a recursively defined sequence, use the given formulas to calculate each term based on the preceding terms. The sequence begins with 1, 1 and follows the pattern s_{n}=2s_{n-1}-2s_{n-2}. The first 12 terms are determined to be 1, 1, 0, -2, -4, -4, 0, 8, 16, 16, 0, -32.

Step-by-step explanation:

The student needs to find the first 12 terms of a sequence defined recursively with s_{1}=1, s_{2}=1, and for n ≥ 3, s_{n}=2s_{n-1}-2s_{n-2}.

1. s_{1} = 1 (given)
2. s_{2} = 1 (given)
3. s_{3} = 2 × s_{2} - 2 × s_{1} = 2 × 1 - 2 × 1 = 0
4. s_{4} = 2 × s_{3} - 2 × s_{2} = 2 × 0 - 2 × 1 = -2
5. s_{5} = 2 × s_{4} - 2 × s_{3} = 2 × (-2) - 2 × 0 = -4
6. s_{6} = 2 × s_{5} - 2 × s_{4} = 2 × (-4) - 2 × (-2) = -8 + 4 = -4
7. s_{7} = 2 × s_{6} - 2 × s_{5} = 2 × (-4) - 2 × (-4) = -8 + 8 = 0
8. s_{8} = 2 × s_{7} - 2 × s_{6} = 2 × 0 - 2 × (-4) = 0 + 8 = 8
9. s_{9} = 2 × s_{8} - 2 × s_{7} = 2 × 8 - 2 × 0 = 16
10. s_{10} = 2 × s_{9} - 2 × s_{8} = 2 × 16 - 2 × 8 = 32 - 16 = 16
11. s_{11} = 2 × s_{10} - 2 × s_{9} = 2 × 16 - 2 × 16 = 32 - 32 = 0
12. s_{12} = 2 × s_{11} - 2 × s_{10} = 2 × 0 - 2 × 16 = -32

Therefore, the first 12 terms of the sequence are 1, 1, 0, -2, -4, -4, 0, 8, 16, 16, 0, -32.