Question 1
Given the sequence
31, 61, 91, 121,...
An arithmetic sequence has a constant difference 'd' and is defined by
[tex]a_n=a_1+\left(n-1\right)d[/tex]
computing the differences of all the adjacent terms
61 - 31 = 30, 91 - 61 = 30, 121 - 91 = 30
The difference between all the adjacent terms is the same and equal to
d = 30
As the first term of the sequence is:
a₁ = 31
now substituting a₁ = 31 and d = 30 in the nth term of the sequence
[tex]a_n=a_1+\left(n-1\right)d[/tex]
[tex]a_n=30\left(n-1\right)+31[/tex]
[tex]a_n=30n+1[/tex]
Now, putting n = 36 to determine the 36th term
[tex]a_n=30n+1[/tex]
[tex]a_{36}=30\left(36\right)+1[/tex]
[tex]a_{36}=1080+1[/tex]
[tex]a_{36}=1081[/tex]
Thus, the 36th term is:
[tex]a_{36}=1081[/tex]
Question 2
Given the sequence
-34, -44, -54, -64, ...
An arithmetic sequence has a constant difference 'd' and is defined by
[tex]a_n=a_1+\left(n-1\right)d[/tex]
computing the differences of all the adjacent terms
[tex]-44-\left(-34\right)=-10,\:\quad \:-54-\left(-44\right)=-10,\:\quad \:-64-\left(-54\right)=-10[/tex]
The difference between all the adjacent terms is the same and equal to
[tex]d=-10[/tex]
As the first term of the sequence is:
a₁ = -34
now substituting a₁ = -34 and d = -10 in the nth term of the sequence
[tex]a_n=a_1+\left(n-1\right)d[/tex]
[tex]a_n=-10\left(n-1\right)-34[/tex]
[tex]a_n=-10n-24[/tex]
Now, putting n = 26 to determine the 36th term
[tex]a_n=-10n-24[/tex]
[tex]a_{26}=-10\left(26\right)-24[/tex]
[tex]a_{26}=-260-24[/tex]
[tex]a_{26}=-284[/tex]
Thus, the 26th term is:
[tex]a_{26}=-284[/tex]
