Problems from Arithmetical progression: definition

Development:

a) Let's see what the difference between the consecutive terms is:

$a_2-a_1={\displaystyle \frac{3}{2}}-1={\displaystyle \frac{1}{2}}$

$a_3-a_2={\displaystyle \frac{5}{2}}-{\displaystyle \frac{3}{2}}={\displaystyle \frac{2}{2}}=1$

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:

$a_2-a_1={\displaystyle \frac{6}{5}}-1={\displaystyle \frac{1}{5}}$

$a_3-a_2={\displaystyle \frac{7}{5}}-{\displaystyle \frac{6}{5}}={\displaystyle \frac{1}{5}}$

$a_4-a_3={\displaystyle \frac{8}{5}}-{\displaystyle \frac{7}{5}}={\displaystyle \frac{1}{5}}$

$a_5-a_4={\displaystyle \frac{9}{5}}-{\displaystyle \frac{8}{5}}={\displaystyle \frac{1}{5}}$

$a_6-a_5=2-{\displaystyle \frac{9}{5}}={\displaystyle \frac{1}{5}}$

As such, we can conclude that is an arithmetical progression with difference $d={\displaystyle \frac{1}{5}}$.

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 $d={\displaystyle \frac{1}{5}}$ c) It is an arithmetical progression with difference $d=3$

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