Answer:
Number Sequence: a set of numbers that follow a pattern or a rule, where each number in the sequence is called a term.
Arithmetic Sequence: has a constant difference between each term, so the difference between each term is the same.
Geometric Sequence: has a constant ratio (multiplier) between each term, so each term is multiplied by the same number.
To determine the type of sequence, calculate the differences between the terms:
[tex]7.5 \underset{-3.75}{\longrightarrow} 3.75 \underset{-1.875}{\longrightarrow} 1.875[/tex]
Therefore, this is not an arithmetic sequence, as the difference between the terms is not the same.
General form of a geometric sequence:
[tex]a_n=ar^{n-1}[/tex]
(where a is the first term and r is the common ratio)
To find the common ratio r, divide consecutive terms:
[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{3.75}{7.5}=0.5[/tex]
Therefore:
[tex]a_n=7.5(0.5)^{n-1}[/tex]
Finding the 4th and 5th terms:
[tex]\implies a_4=7.5(0.5)^{4-1}=0.9375[/tex]
[tex]\implies a_5=7.5(0.5)^{5-1}=0.46875[/tex]
Part (a)
[tex]\large \begin{array}{| c | c | c | c | c | c |}\cline{1-6} n & 1 & 2 & 3 & 4 & 5 \\\cline{1-6} t(n) & 7.5 & 3.75 & 1.875 & 0.9375 & 0.46875 \\\cline{1-6}\end{array}[/tex]
Part (b)
Geometric sequence
Part (c)
[tex]a_n=7.5(0.5)^{n-1}[/tex]
