Question

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

(a) List out at least the first 4 terms of the sequences. Be sure to use proper notation.

(b) Give the closed/explicit formula for this sequence. Be sure to use proper notation.

(c) Give the recursive formula for this sequence. Be sure to use proper notation.

Answer 1

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.`

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)