Q:

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

Accepted Solution

A:
Answer:Part A[tex]f(n)=52-12(n-1)[/tex][tex]f(n)=\left\{\begin{matrix}52\: \:if \: \:n=1 & \\f(n+1)+12& if\: n\geq 2 \end{matrix}\right.[/tex]Part B[tex](2,4,8,16,32)\: \:[/tex] Geometric SequencePart C1/4,3/4,5/4,7/4,9/4[tex]g(n)=\frac{1}{4}+\frac{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[/tex]Part D:[tex]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[/tex]Step-by-step explanation:By definition, an Arithmetic Sequence holds the same difference between each following number.Part A[tex](52,40, 28, 16)\\52-40=12\\40-28=12\\28-16=12\\d=12[/tex]Explicit FormulaTo write an explicit formula is to write it as function.[tex]f(n)=52-12(n-1)[/tex]Recursive FormulaTo write it as recursive formula, is to write it as recurrence given to some restrictions: [tex]f(n)=\left\{\begin{matrix}52\: \:if \: \:n=1 & \\f(n+1)+12& if\: n\geq 2 \end{matrix}\right.[/tex]Part B[tex](2,4,8,16,32)\: \:[/tex]Geometric Sequence, since 2*2=4 8*2=16 and 16*2=32 and 8+2=10 8+16=24Part C[tex](\frac{1}{4},\frac{3}{4},\frac{5}{4},\frac{7}{4},\frac{9}{4})\\\[/tex]Arithmetic Sequence, difference[tex]d=\frac{2}{4}[/tex]Explicit Formula:[tex]g(n)=\frac{1}{4}+\frac{2}{4}(n-1)[/tex]Recursive Formula[tex]g(n)=\left\{\begin{matrix}\frac{1}{4} &if\:n=1 \\ g(n+1)+\frac{2}{4} &if\: n\geq 2\end{matrix}\right.[/tex]Part D(1.1,1.5,1.9,2.3,2.7) Arithmetic Sequence, difference d=0.4Explicit formula[tex]h(n)=1.1+0.4(n-1)\\[/tex]Recursive Formula[tex]h(n)=\left\{\begin{matrix}1.1 &if\:n=1 \\ h(n+1)+0.4 &if\: n\geq 2\end{matrix}\right.[/tex]