Answer:
a) The explicit formula is [tex]a_n=7n+5[/tex]
b) The 31st term of the sequence is 222
Step-by-step explanation:
a) The explicit formula for an arithmetic sequence is:
[tex]a_n=a_1+(n-1)d[/tex]
The recursive formula for the sequence is
[tex]a_1=12\\a_n=a_{n-1}+7[/tex]
From this formula we can find the common difference by comparing to the general recursive formula:[tex]\\a_n=a_{n-1}+d[/tex]
This means the common difference d=7.
We now substitute the first term [tex]a_1=12[/tex] and the common difference [tex]d=7[/tex] into the explicit formula: [tex]a_n=a_1+(n-1)d[/tex] to obtain:
[tex]a_n=12+(n-1)\times 7[/tex]
Expand to get:
[tex]a_n=12+7n-7[/tex]
Simplify:
[tex]a_n=7n+5[/tex]
b) To find the 31st term of this sequence, we substitute n=31 in to the the explicit formula, [tex]a_n=7n+5[/tex] to obtain:
[tex]a_{31}=7*31+5[/tex]
Multiply to get:
[tex]a_{31}=217+5[/tex]
Add to get:
[tex]a_{31}=222[/tex]
Therefore the 31st term of the sequence is 222
