Question

Are the following sequences arithmetic? If so, what is the common difference and the explicit formula?
7) 183, 1835, 18355, 183555

8) -1, 1, 4, 8

9) -35, -42, -49, -56

10) 9, 0, -9, -18

11) 9, 17, 25, 33

12) 2, 5, 10, 17

Answer 1

Are the following sequences arithmetic? If so, what is the common difference and the explicit formula?

1) 183, 1835, 18355, 183555

$$\mathrm{d}_1 = 1835 - 183 = 1652$$ $$\mathrm{d}_2 = 18355 - 1835 = 16520$$ $$\mathrm{d}_1 \neq \mathrm{d}_2$$

So it is not an arithmetic sequence

2) $-1, 1, 4, 8$

$$\mathrm{d}_1 = 1 - (-1) = 2$$ $$\mathrm{d}_2 = 4 - 1 = 3$$ $$\mathrm{d}_1 \neq \mathrm{d}_2$$

So it is not an arithmetic sequence

3) $-35, -42, -49, -56$

$$\mathrm{d}_1 = -42 - (-35) = -7$$ $$\mathrm{d}_2 = -49 - (-42) = -7$$ $$\mathrm{d}_3 = -56 - (-49) = -7$$ $$\mathrm{d}_1 = \mathrm{d}_2 = \mathrm{d}_3 = \mathrm{d}$$

So it is arithmetic sequence $a_1 = -35$, $d = -7$, $a_n = a_{n-1} - 7$

4) $9, 0, -9, -18$

$$\mathrm{d}_1 = 0 - 9 = -9$$ $$\mathrm{d}_2 = -9 - 0 = -9$$ $$\mathrm{d}_3 = -18 - (-9) = -9$$ $$\mathrm{d}_1 = \mathrm{d}_2 = \mathrm{d}_3 = \mathrm{d}$$

So this is arithmetic sequence $a_1 = 9$, $d = -9$, $a_n = a_{n-1} - 9$

5) $9, 17, 25, 33$

$$\mathrm{d}_1 = 17 - 9 = 8$$ $$\mathrm{d}_2 = 25 - 17 = 8$$ $$\mathrm{d}_3 = 33 - 25 = 8$$ $$\mathrm{d}_1 = \mathrm{d}_2 = \mathrm{d}_3 = \mathrm{d}$$

So it is arithmetic sequences $a_1 = 9$, $d = 8$, $a_n = a_{n-1} + 8$

6) 2,5,10,17

$$\mathrm{d}_1 = 5 - 2 = 3$$ $$\mathrm{d}_2 = 10 - 5 = 5$$ $$\mathrm{d}_1 \neq \mathrm{d}_2$$

So this is not an arithmetic sequence