Answer:
A) The first 4 terms of the sequences are: [tex]a_{1} =16[/tex], [tex]a_{2} =24[/tex], [tex]a_{3} =32[/tex] and [tex]a_{4} =40[/tex].
B) An explicit formula for this sequence can be written as: [tex]a_{n} =8*(n+1)[/tex]
C) A recursive formula for this sequence can be written as:
[tex]\left \{ {{a_{1} =16} \atop {a_{n} =a_{n-1}+8}} \right.[/tex]
Step-by-step explanation:
A) You can find the firs terms of this sequence simply selecting an odd integer and summing the consecutive 3 ones:
[tex]a_{n} = Odd_{n}+Odd_{n+1}+Odd_{n+2}+Odd_{n+3}[/tex] (a.1)
[tex]a_{1}=1+3+5+7=16[/tex]
[tex]a_{2}=3+5+7+9=24[/tex]
[tex]a_{3}=5+7+9+11=32[/tex]
[tex]a_{4}=7+9+11+13=40[/tex]
B) Observe the sequence of odd numbers 1, 3, 5, 7, 9, 11, 13(...).
You can express this sequence as:
[tex]Odd_{n}=(2*n-1)[/tex] (b.1)
If you merge the expression b.1 in a.1, you obtain the explicit formula of the sequence:
[tex]a_{n} = Odd_{n}+Odd_{n+1}+Odd_{n+2}+Odd_{n+3}[/tex] (a.1)
[tex]a_{n} = (2*n-1)+((2*(n+1)-1))+((2*(n+2)-1))+((2*(n+3)-1))[/tex] (b.2)
[tex]a_{n} = 8*n+8[/tex] (b.3)
[tex]a_{n} =8*(n+1)[/tex] (b.s)
C) The recursive formula has to be written considering an initial term and an N term linked with the previous term. You can see an addition of 8 between a term and the next one. So you can express each term as an addition of 8 with the previous one. Therefore, if the first term is 16:
[tex]\left \{ {{a_{1} =16} \atop {a_{n} =a_{n-1}+8}} \right.[/tex] (c.s)
