Answer:
a) The error is that, the initial value is n=1 NOT n=3
b) The sum is [tex]a_n=a_{n+1}+5=192[/tex]
c)The explicit formula is [tex]a_n=5n+3[/tex]
The recursive formula is [tex]a_n=a_{n+1}+5[/tex],
Step-by-step explanation:
The given arithmetic series is 8 + 13 + ... + 43.
The first term is [tex]a_1=8[/tex], the common difference is [tex]d=13-8=5[/tex]
The nth term is given by:
[tex]a_n=a_1+d(n-1)[/tex]
We substitute the values to get:
[tex]a_n=8+5(n-1)\\a_n=8+5n-5\\\\a_n=3+5n[/tex]
To find how many terms are in the sequence we solve the equation:
[tex]3+5n=43\\\implies 5n=43-3\\5n=40\\n=8[/tex]
The summation notation is [tex]\sum_{n=1}^8(3+5n)[/tex]
The error the student made is in the initial value.
It should be n=1 NOT n=3
b) The sum of the arithmetic series is calculated using:
[tex]S_n=\frac{n}{2}(a+l)[/tex]
We substitute o get:
[tex]S_8=\frac{8}{2}(5+43)[/tex]
[tex]S_8=4(48)[/tex]
[tex]S_8=192[/tex]
c) The explicit formula we already calculated in a), which is [tex]a_n=3+5n[/tex]
The recursive formula is given as:
[tex]a_n=a_{n+1}+d[/tex]
We substitute d=5 to get:
[tex]a_n=a_{n+1}+5[/tex]
