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Write an expression for the apparent nth term a_n of the sequence. (Assume that n begins with 1.)

a. 1, -1, 1, -1, 1, . . .

a_n =

b. -1, 2, 5, 8, 11, . . .
a_n =


a_n=

Answer:

[tex]\textsf{a)} \quad a_n=(-1)^{n-1}[/tex]

[tex]\textsf{b)} \quad a_n=3n-4[/tex]

Step-by-step explanation:

Given sequence:

1, -1, 1, -1, 1, ...

The given sequence is geometric since there is a common ratio of -1 between consecutive terms.

To find the common ratio of a geometric sequence, divide a term by the previous term:

[tex]\dfrac{a_5}{a_4}=\dfrac{1}{-1}=-1[/tex]

[tex]\dfrac{a_4}{a_3}=\dfrac{-1}{1}=-1[/tex]

[tex]\dfrac{a_3}{a_2}=\dfrac{1}{-1}=-1[/tex]

[tex]\dfrac{a_2}{a_1}=\dfrac{-1}{1}=-1[/tex]

[tex]\boxed{\begin{minipage}{5.5 cm}\underline{Geometric sequence}\\\\$a_n=ar^{n-1}$\\\\where:\\\phantom{ww}$\bullet$ $a$ is the first term. \\\phantom{ww}$\bullet$ $r$ is the common ratio.\\\phantom{ww}$\bullet$ $a_n$ is the $n$th term.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given:

Substitute the values of a and r into the formula to create an equation for the nth term:

[tex]\implies a_n=1 \cdot (-1)^{n-1}[/tex]

[tex]\implies a_n=(-1)^{n-1}[/tex]

Given sequence:

-1, 2, 5, 8, 11, ...

The given sequence is arithmetic since there is a common difference of 3 between consecutive terms.

To find the common difference of an arithmetic sequence, subtract consecutive terms:

[tex]a_2-a_1=2-(-1)=3[/tex]

[tex]a_3-a_2=5-2=3[/tex]

[tex]a_4-a_3=8-5=3[/tex]

[tex]a_5-a_4=11-8=3[/tex]

[tex]\boxed{\begin{minipage}{8 cm}\underline{General form of an arithmetic sequence}\\\\$a_n=a+(n-1)d$\\\\where:\\\phantom{ww}$\bullet$ $a_n$ is the nth term. \\ \phantom{ww}$\bullet$ $a$ is the first term.\\\phantom{ww}$\bullet$ $d$ is the common difference between terms.\\\phantom{ww}$\bullet$ $n$ is the position of the term.\\\end{minipage}}[/tex]

Given:

Substitute the values of a and d into the formula to create an equation for the nth term:

[tex]\implies a_n=-1+(n-1)3[/tex]

[tex]\implies a_n=-1+3n-3[/tex]

[tex]\implies a_n=3n-4[/tex]

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