Answer:
a) See below.
b) Arithmetic sequence
[tex]\textsf{c)} \quad a_n=10n-7[/tex]
Step-by-step explanation:
Part (a)
From inspection of the given table, t(n) increases by 10 each time n increases by 1.
Therefore:
[tex]\implies a_7=53+10=63[/tex]
[tex]\implies a_8=63+10=73[/tex]
Completed table:
[tex]\begin{array}{|c|c|}\cline{1-2} \vphantom{\dfrac12} n&t(n) \\\cline{1-2} \vphantom{\dfrac12} 4& 33\\\cline{1-2} \vphantom{\dfrac12} 5& 43\\\cline{1-2} \vphantom{\dfrac12} 6&53 \\\cline{1-2} \vphantom{\dfrac12} 7&63\\\cline{1-2} \vphantom{\dfrac12} 8& 73\\\cline{1-2} \end{array}[/tex]
Part (b)
As the given sequence has a constant difference of 10, it is an arithmetic sequence.
Part (c)
[tex]\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}}[/tex]
The common difference is 10. Therefore:
To find the first term, a, substitute the value of d and one of the terms into the formula:
[tex]\begin{aligned}\implies a_4=a+(4-1)(10)&=33\\a+3(10)&=33\\a+30&=33\\&a=3\end{aligned}[/tex]
Therefore, to write an equation for the given arithmetic sequence, substitute the found values of a and d into the formula:
[tex]\implies a_n=3+(n-1)(10)[/tex]
[tex]\implies a_n=3+10n-10[/tex]
[tex]\implies a_n=10n-7[/tex]
