Question

Let an be the sum of the first n positive odd integers.

Answer 1

Answer:

A) The first 4 terms of the sequences are: $a_{1} =16$, $a_{2} =24$, $a_{3} =32$ and $a_{4} =40$.

B) An explicit formula for this sequence can be written as: $a_{n} =8*(n+1)$

C) A recursive formula for this sequence can be written as:

$\left \{ {{a_{1} =16} \atop {a_{n} =a_{n-1}+8}} \right.$

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:

$a_{n} = Odd_{n}+Odd_{n+1}+Odd_{n+2}+Odd_{n+3}$ (a.1)

$a_{1}=1+3+5+7=16$

$a_{2}=3+5+7+9=24$

$a_{3}=5+7+9+11=32$

$a_{4}=7+9+11+13=40$

B) Observe the sequence of odd numbers 1, 3, 5, 7, 9, 11, 13(...).

You can express this sequence as:

$Odd_{n}=(2*n-1)$ (b.1)

If you merge the expression b.1 in a.1, you obtain the explicit formula of the sequence:

$a_{n} = Odd_{n}+Odd_{n+1}+Odd_{n+2}+Odd_{n+3}$ (a.1)

$a_{n} = (2*n-1)+((2*(n+1)-1))+((2*(n+2)-1))+((2*(n+3)-1))$ (b.2)

$a_{n} = 8*n+8$ (b.3)

$a_{n} =8*(n+1)$ (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:

$\left \{ {{a_{1} =16} \atop {a_{n} =a_{n-1}+8}} \right.$ (c.s)