Answer:
a) See below.
b) Geometric sequence
[tex]\textsf{c)} \quad a_n=7.5(0.5)^{n-1}[/tex]
Step-by-step explanation:
Part (a)
From inspection of the given table, t(n) halves each time n increases by 1.
Therefore:
[tex]\implies a_4=1.875 \div 2 = 0.9375[/tex]
[tex]\implies a_5=0.9375 \div 2 =0.46875[/tex]
Completed table:
[tex]\begin{array}{|c|c|}\cline{1-2} \vphantom{\dfrac12} n&t(n) \\\cline{1-2} \vphantom{\dfrac12} 1& 7.5\\\cline{1-2} \vphantom{\dfrac12} 2& 3.75\\\cline{1-2} \vphantom{\dfrac12} 3&1.875 \\\cline{1-2} \vphantom{\dfrac12} 4& 0.9375\\\cline{1-2} \vphantom{\dfrac12} 5& 0.46875\\\cline{1-2} \end{array}[/tex]
Part (b)
As the given sequence has a constant ratio of 0.5, it is a geometric sequence.
Part (c)
[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]
The first term of the sequence is 7.5. Therefore,:
The common ratio is 0.5. Therefore:
Therefore, to write an equation for the given geometric sequence, substitute the found values of a and r into the formula:
- [tex]a_n=7.5(0.5)^{n-1}[/tex]
