1. Problem: List the first 10 terms of each sequence and identify if any are arithmetic or geometric progressions. 2. a) Sequence starting at 10, subtracting 3 each time: Formula: $a_n = a_{n-1} - 3$, with $a_1 = 10$ Terms: 10, 7, 4, 1, -2, -5, -8, -11, -14, -17 This is an arithmetic progression with common difference $-3$. 3. b) Sequence where $n$th term is sum of first $n$ positive integers: Formula: $a_n = \frac{n(n+1)}{2}$ Terms: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55 This is neither arithmetic nor geometric. 4. c) Sequence where $n$th term is $\lfloor \sqrt{n} \rfloor$: Terms: 1, 1, 1, 2, 2, 2, 2, 2, 3, 3 Not arithmetic or geometric. 5. d) Sequence with first two terms 1 and 5, each next term sum of previous two: Formula: $a_n = a_{n-1} + a_{n-2}$, $a_1=1$, $a_2=5$ Terms: 1, 5, 6, 11, 17, 28, 45, 73, 118, 191 Not arithmetic or geometric. 6. e) Sequence constructed: start 1, then add 1, multiply by 1, add 2, multiply by 2, etc. Steps: Term 1: 1 Term 2: 1 + 1 = 2 Term 3: 2 * 1 = 2 Term 4: 2 + 2 = 4 Term 5: 4 * 2 = 8 Term 6: 8 + 3 = 11 Term 7: 11 * 3 = 33 Term 8: 33 + 4 = 37 Term 9: 37 * 4 = 148 Term 10: 148 + 5 = 153 Terms: 1, 2, 2, 4, 8, 11, 33, 37, 148, 153 Not arithmetic or geometric. 7. f) Sequence where $a_n$ is largest integer $k$ with $k! \leq n$: Terms: 1, 1, 2, 2, 2, 3, 3, 4, 4, 4 Not arithmetic or geometric. 8. Exercise 2: 9. a) Recurrence relation for salary $S_n$ years after 2025: $S_0 = 120000$ $S_n = 1.05 S_{n-1} + 2000$ 10. b) Salary in 2028 ($n=3$): Calculate stepwise: $S_1 = 1.05 \times 120000 + 2000 = 126000 + 2000 = 128000$ $S_2 = 1.05 \times 128000 + 2000 = 134400 + 2000 = 136400$ $S_3 = 1.05 \times 136400 + 2000 = 143220 + 2000 = 145220$ 11. c) Explicit formula: General form for $S_n = r S_{n-1} + d$ with $S_0 = S$ is $$S_n = r^n S + d \frac{r^n - 1}{r - 1}$$ Here, $r=1.05$, $d=2000$, $S=120000$ So, $$S_n = 120000 \times (1.05)^n + 2000 \times \frac{(1.05)^n - 1}{0.05}$$ Final answers: - First 10 terms of each sequence as above. - Only sequence (a) is arithmetic progression. - Salary recurrence and explicit formula as above. - Salary in 2028 is 145220.