1. **Identify arithmetic sequences and find $t_1$, $d$, and next three terms** An arithmetic sequence has a constant difference $d$ between consecutive terms. (a) Sequence: 16, 32, 48, 64, 80, ... - Calculate $d = 32 - 16 = 16$ - $t_1 = 16$ - Next three terms: $80 + 16 = 96$, $96 + 16 = 112$, $112 + 16 = 128$ (b) Sequence: 2, 4, 8, 16, 32, ... - Differences: $4-2=2$, $8-4=4$, $16-8=8$ (not constant) - Not arithmetic (c) Sequence: -4, -7, -10, -13, -16, ... - Calculate $d = -7 - (-4) = -3$ - $t_1 = -4$ - Next three terms: $-16 + (-3) = -19$, $-19 + (-3) = -22$, $-22 + (-3) = -25$ 2. **Find indicated terms using $t_n = t_1 + (n-1)d$ or given formula** (a) $t_n = 3n - 7$, find $t_8$ - Substitute $n=8$: $t_8 = 3(8) - 7 = 24 - 7 = 17$ (b) $t_n = -\frac{2}{3} n + 12$, find $t_6$ - Substitute $n=6$: $t_6 = -\frac{2}{3} \times 6 + 12 = -4 + 12 = 8$ 3. **Find number of terms in arithmetic sequences** Use formula for $n$th term: $t_n = t_1 + (n-1)d$ (a) Sequence: 5, 2, ..., -43 - $t_1 = 5$, $d = 2 - 5 = -3$, $t_n = -43$ - Solve for $n$: $-43 = 5 + (n-1)(-3)$ - $-43 - 5 = (n-1)(-3)$ - $-48 = -3(n-1)$ - Divide both sides by $-3$: $$-48 = \cancel{-3}(n-1) \Rightarrow \frac{-48}{\cancel{-3}} = n-1 \Rightarrow 16 = n-1$$ - $n = 17$ (b) Sequence: -8, -11, -14, ..., -95 - $t_1 = -8$, $d = -11 - (-8) = -3$, $t_n = -95$ - $-95 = -8 + (n-1)(-3)$ - $-95 + 8 = (n-1)(-3)$ - $-87 = -3(n-1)$ - Divide both sides by $-3$: $$-87 = \cancel{-3}(n-1) \Rightarrow \frac{-87}{\cancel{-3}} = n-1 \Rightarrow 29 = n-1$$ - $n = 30$ (c) Sequence: 17.5, 16.25, 15, ..., -51.25 - $t_1 = 17.5$, $d = 16.25 - 17.5 = -1.25$, $t_n = -51.25$ - $-51.25 = 17.5 + (n-1)(-1.25)$ - $-51.25 - 17.5 = (n-1)(-1.25)$ - $-68.75 = -1.25(n-1)$ - Divide both sides by $-1.25$: $$-68.75 = \cancel{-1.25}(n-1) \Rightarrow \frac{-68.75}{\cancel{-1.25}} = n-1 \Rightarrow 55 = n-1$$ - $n = 56$ 4. **Find given terms using term formula $t_n = t_1 + (n-1)d$** (a) Sequence: 4, 15, 26, ... - $t_1 = 4$, $d = 15 - 4 = 11$ - Find $t_{53}$: $$t_{53} = 4 + (53-1) \times 11 = 4 + 52 \times 11 = 4 + 572 = 576$$ (b) Sequence: -33, -22, -11, ... - $t_1 = -33$, $d = -22 - (-33) = 11$ - Find $t_{43}$: $$t_{43} = -33 + (43-1) \times 11 = -33 + 42 \times 11 = -33 + 462 = 429$$ 5. **Graph of arithmetic sequence with points (1,5), (2,10), ..., (12,60)** (a) First five terms: 5, 10, 15, 20, 25 (b) General term formula: $t_n = t_1 + (n-1)d$ - $t_1 = 5$, $d = 10 - 5 = 5$ - So, $t_n = 5 + (n-1)5 = 5 + 5n - 5 = 5n$ (c) Find $t_{50}$ and $t_{200}$: - $t_{50} = 5 \times 50 = 250$ - $t_{200} = 5 \times 200 = 1000$ (d) The slope of the graph is the common difference $d = 5$, which is the coefficient of $n$ in the formula. (e) The y-intercept corresponds to the value when $n=0$, which is 0 here, matching the formula $t_n = 5n$. 6. **Find first term when $t_{16} = 110$ and $d=7$** Use $t_n = t_1 + (n-1)d$ - $110 = t_1 + (16-1)7$ - $110 = t_1 + 15 imes 7 = t_1 + 105$ - $t_1 = 110 - 105 = 5$ 7. **Find $t_1$ and general term given $t_7 = 9$ and $t_{34} = -72$** Use $t_n = t_1 + (n-1)d$ - From $t_7$: $9 = t_1 + 6d$ - From $t_{34}$: $-72 = t_1 + 33d$ - Subtract equations: $(-72) - 9 = (t_1 + 33d) - (t_1 + 6d)$ $-81 = 27d$ - Solve for $d$: $d = \frac{-81}{27} = -3$ - Substitute $d$ back: $9 = t_1 + 6(-3) = t_1 - 18$ - $t_1 = 9 + 18 = 27$ - General term: $t_n = 27 + (n-1)(-3) = 27 - 3n + 3 = 30 - 3n$ 8. **Kamra's push-ups: $t_5 = 9$, $t_{15} = 29$** (a) Find general term $t_n = t_1 + (n-1)d$ - From $t_5$: $9 = t_1 + 4d$ - From $t_{15}$: $29 = t_1 + 14d$ - Subtract: $29 - 9 = (t_1 + 14d) - (t_1 + 4d)$ $20 = 10d$ - $d = 2$ - Substitute $d$ back: $9 = t_1 + 4 imes 2 = t_1 + 8$ - $t_1 = 1$ - General term: $t_n = 1 + (n-1)2 = 2n - 1$ (b) Find day $n$ when $t_n = 75$ - $75 = 2n - 1$ - $75 + 1 = 2n$ - $76 = 2n$ - $n = 38$ **Final answers:** - Arithmetic sequences: (a) and (c) - $t_8 = 17$, $t_6 = 8$ - Number of terms: (a) 17, (b) 30, (c) 56 - $t_{53} = 576$, $t_{43} = 429$ - Graph sequence formula: $t_n = 5n$ - First term for problem 6: 5 - Problem 7: $t_1 = 27$, $t_n = 30 - 3n$ - Kamra's push-ups: $t_n = 2n - 1$, day for 75 push-ups is 38