Answer:
Common difference: 30
General term formula: [tex]a_n=30n-9[/tex]
52nd term: 1551
Step-by-step explanation:
Given sequence:
21, 51, 81, 111
Calculate the difference in terms:
[tex]21 \underset{+30}{\longrightarrow} 51 \underset{+30}{\longrightarrow} 81 \underset{+30}{\longrightarrow} 111[/tex]
The sequence is increasing by 30 each time, so as there is a common difference, this is an arithmetic sequence with a common difference of 30.
General form of an arithmetic sequence
[tex]a_n=a+(n-1)d[/tex]
where:
Given:
Substitute the given values into the formula to find the general term formula:
[tex]\implies a_n=21+(n-1)30[/tex]
[tex]\implies a_n=21+30n-30[/tex]
[tex]\implies a_n=30n-9[/tex]
To find the 52nd term, simply substitute n = 52 into the found general term formula:
[tex]\implies a_{52}=30(52)-9[/tex]
[tex]\implies a_{52}=1560-9[/tex]
[tex]\implies a_{52}=1551[/tex]
