meredith48034
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a. Identify each sequence as arithmetic, geometric, or neither.
b. If it is arithmetic or geometric, describe the sequence generator

n t(n)
0 16
1 8
2 4
3 2
4 1
5 1/2
6 1/4



Answer :

mhanifa

Answer:

  • a) The sequence is geometric,
  • b) [tex]t(n) = 2^{4-n}[/tex]

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Given sequence from zeros to 6th terms:

  • 16, 8, 4, 2, 1, 1/2, 1/4

It is a GP with the common ratio of 1/2 as each term is half the previous one.

Use the nth term equation, considering the first term is t(1) = 8, and common ratio r = 1/2:

  • [tex]t(n) = t(1)*r^{n-1}[/tex]
  • [tex]t(n) = 8*(1/2)^{n-1}=8*2^{1-n}=2^3*2^{1-n}=2^{4-n}[/tex]
semsee45

Answer:

a)  Geometric sequence

[tex]\textsf{b)} \quad t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]

Step-by-step explanation:

An Arithmetic Sequence has a constant difference between each consecutive term.

A Geometric Sequence has a constant ratio (multiplier) between each consecutive term.

Part (a)

From inspection of the given table, t(n) halves each time n increases by 1.

Therefore, the sequence is geometric with a common ratio of 1/2.

Part (b)

[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:

  • a = 8
  • r = 1/2
  • aₙ = t(n)

Substitute the initial value (when n = 1) and the common ratio into the formula to create an equation for the nth term:

[tex]\implies t(n)=8 \left(\dfrac{1}{2}\right)^{n-1}[/tex]

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