Question

NO LINKS!!!

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 1

`\textsf{a)} \quad a_n=(-1)^(n-1)`

`\textsf{b)} \quad a_n=3n-4`

Explanation:

Sequence a

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:

`(a_5)/(a_4)=(1)/(-1)=-1`

`(a_4)/(a_3)=(-1)/(1)=-1`

`(a_3)/(a_2)=(1)/(-1)=-1`

`(a_2)/(a_1)=(-1)/(1)=-1`

`\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}}`

Given:

• a = 1
• r = -1

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

`\implies a_n=1 \cdot (-1)^(n-1)`

`\implies a_n=(-1)^(n-1)`

Sequence b

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:

`a_2-a_1=2-(-1)=3`

`a_3-a_2=5-2=3`

`a_4-a_3=8-5=3`

`a_5-a_4=11-8=3`

`\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}}`

Given:

• a = -1
• d = 3

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

`\implies a_n=-1+(n-1)3`

`\implies a_n=-1+3n-3`

`\implies a_n=3n-4`