Answer:
To determine the type of sequence, calculate the differences between the terms:
[tex]33 \underset{+10}{\longrightarrow} 43 \underset{+10}{\longrightarrow} 53[/tex]
Therefore, this is an arithmetic sequence, as the difference between the terms is constant → the common difference is 10.
General form of an arithmetic sequence:
[tex]a_n=a+(n-1)d[/tex]
where:
- [tex]a_n[/tex] is the nth term
- a is the first term
- d is the common difference between terms
To find the first term, substitute the known values into the formula and solve for a:
[tex]\begin{aligned}\implies a_4=a+(4-1)10 & =33\\a+30 & = 33\\a & = 3\end{aligned}[/tex]
Therefore:
[tex]\implies a_n=3+(n-1)10[/tex]
[tex]\implies a_n=3+10n-10[/tex]
[tex]\implies a_n=10n-7[/tex]
Finding the 7th and 8th terms:
[tex]\implies a_7=10(7)-7=63[/tex]
[tex]\implies a_8=10(8)-7=73[/tex]
Part (a)
[tex]\large \begin{array}{| c | c | c | c | c | c |}\cline{1-6} n & 4 & 5 & 6 & 7 & 8 \\\cline{1-6} t(n) & 33 & 43 & 53 & 63 & 73 \\\cline{1-6}\end{array}[/tex]
Part (b)
Arithmetic sequence
Part (c)
[tex]a_n=10n-7[/tex]
