Question

State whether each sequence is arithmetic and justify your answer. If the sequence is arithmetic, write a recursive and an explicit formula to represent it. Part A: 52,40,28,16 Part B: 2,4,8,16,32 Part C: 1/4,3/4,5/4,7/4,9/4 Part D: 1.1,1.5,1.9,2.3,2.7

Answer 1

Part A

`f(n)=52-12(n-1)`

`f(n)=\left\{\begin{matrix}52\: \:if \: \:n=1 & \\f(n+1)+12& if\: n\geq 2 \end{matrix}\right.`

Part B

`(2,4,8,16,32)\: \:` Geometric Sequence

Part C

1/4,3/4,5/4,7/4,9/4

`g(n)=(1)/(4)+(2)/(4)(n-1)\\f(n)=\left\{\begin{matrix}1/4if \: \:n=1 & \\ f(n+1)+2/4& if\: n\geq 2 \end{matrix}\right`

Part D:

`h(n)=1.1+0.4(n-1)\\h(n)=\left\{\begin{matrix}1.1 & if\:n=1 \\ h(n+1)+0.4 & if\:n\geq 2\end{matrix}\right`

Explanation:

By definition, an Arithmetic Sequence holds the same difference between each following number.

Part A

`(52,40, 28, 16)\\52-40=12\\40-28=12\\28-16=12\\d=12`

Explicit Formula

To write an explicit formula is to write it as function.

`f(n)=52-12(n-1)`

Recursive Formula

To write it as recursive formula, is to write it as recurrence given to some restrictions:

`f(n)=\left\{\begin{matrix}52\: \:if \: \:n=1 & \\f(n+1)+12& if\: n\geq 2 \end{matrix}\right.`

Part B

`(2,4,8,16,32)\: \:`

Geometric Sequence, since 2*2=4 8*2=16 and 16*2=32 and 8+2=10 8+16=24

Part C

`((1)/(4),(3)/(4),(5)/(4),(7)/(4),(9)/(4))\\\`

Arithmetic Sequence, difference

`d=(2)/(4)`

Explicit Formula:

`g(n)=(1)/(4)+(2)/(4)(n-1)`

Recursive Formula

`g(n)=\left\{\begin{matrix}(1)/(4) &if\:n=1 \\ g(n+1)+(2)/(4) &if\: n\geq 2\end{matrix}\right.`

Part D

(1.1,1.5,1.9,2.3,2.7) Arithmetic Sequence, difference d=0.4

Explicit formula

`h(n)=1.1+0.4(n-1)\\`

Recursive Formula

`h(n)=\left\{\begin{matrix}1.1 &if\:n=1 \\ h(n+1)+0.4 &if\: n\geq 2\end{matrix}\right.`