Answer:
The rule for given sequence is: a_n = -27+16n
And the 100th term is: 1573
Step-by-step explanation:
Given sequence is:
-11, 5, 21, 37, 53, ...
Here
[tex]a_1 = -11\\a_2 = 5\\a_3 = 21\\a_4 = 37[/tex]
First of all, we have to find if this is an arithmetic sequence
For that purpose, the common difference has to be found. Common difference, denoted by d, is the difference between consecutive terms of an arithmetic sequence
So,
[tex]d = a_2 -a_1 = 5-(-11) = 5+11 = 16\\d = a_3 -a_2 = 21-5 = 16\\d = a_4-a_3 = 37-21 = 16[/tex]
As the common difference is same, the sequence is an arithmetic sequence
General rule for arithmetic sequence is:
[tex]a_n = a_1+(n-1)d[/tex]
Putting values
[tex]a_n = -11+(n-1)(16)\\a_n = -11+16n-16\\a_n = -27+16n[/tex]
For 100th term,
Putting n=100
[tex]a_{100} = -27+16(100)\\a_{100} = -27+1600 = 1573[/tex]
Hence,
The rule for given sequence is: a_n = -27+16n
And the 100th term is: 1573
