Identify which of the following successions are arithmetical progressions and calculate the difference:
$$a: \Big(1, \dfrac{3}{2}, \dfrac{5}{2}, \dfrac{7}{2}, \dfrac{9}{2}, \ldots \Big)$$
$$b: \Big(1, \dfrac{6}{5}, \dfrac{7}{5}, \dfrac{8}{5}, \dfrac{9}{5}, 2, \ldots \Big)$$
$$c: (-4,-1,2,5,8,\ldots)$$
Development:
a) Let's see what the difference between the consecutive terms is:
$$a_2 - a_1 = \dfrac{3}{2}-1=\dfrac{1}{2}$$
$$a_3 - a_2 = \dfrac{5}{2}-\dfrac{3}{2}=\dfrac{2}{2}=1$$
As these differences are not constant, it cannot be an arithmetical progression.
b) Let's see all what the difference between the consecutive terms is:
$$a_2 - a_1 = \dfrac{6}{5}-1=\dfrac{1}{5}$$
$$a_3 - a_2 = \dfrac{7}{5}-\dfrac{6}{5}=\dfrac{1}{5}$$
$$a_4 - a_3 = \dfrac{8}{5}-\dfrac{7}{5}=\dfrac{1}{5}$$
$$a_5 - a_4 = \dfrac{9}{5}-\dfrac{8}{5}=\dfrac{1}{5}$$
$$a_6 - a_5 = 2-\dfrac{9}{5}=\dfrac{1}{5}$$
As such, we can conclude that is an arithmetical progression with difference $$d=\dfrac{1}{5}$$.
c) Let's see what the difference between the consecutive terms is:
$$a_2-a_1=-1-(-4)=3$$
$$a_3-a_2=2-(-1)=3$$
$$a_4-a_3=5-2=3$$
$$a_5-a_4=8-5=3$$
Then, we can conclude that is an arithmetical progression with difference $$d=3$$.
Solution:
a) It is not an arithmetical progression. b) It is an arithmetical progression with difference $$d=\dfrac{1}{5}$$ c) It is an arithmetical progression with difference $$d=3$$