Answer:
[tex]$ \textbf{a}_{\textbf{17}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{55} $[/tex]
Step-by-step explanation:
The [tex]$ n^{th} $[/tex] term of an arithmetic sequence is given by:
[tex]$ \textbf{a}_{\textbf{n}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{a} \hspace{1mm} \textbf{+} \hspace{1mm} \textbf{(n - 1)d} $[/tex]
where a is the first term of the sequence
and d is the common difference.
We are given the [tex]$ 3^{rd} $[/tex] and the [tex]$ 13^{th} $[/tex] term of the sequence.
We are asked to find the [tex]$ 17^{th} $[/tex] term.
From the formula, we can write
[tex]$ a_3 = a + (3 - 1)d $[/tex]
[tex]$ \implies 13 = a + 2d \hspace{6mm} \hdots (1) $[/tex]
Also, [tex]$ a_{13} = a + (13 - 1)d $[/tex]
[tex]$ \implies 43 = a + 12d \hspace{6mm} \hdots (2) $[/tex]
Now, we solve Equation (1) and (2) for a and d.
Solving we get:
a = 7; d = 3
Therefore, [tex]$ 17^{th} $[/tex] term, [tex]$ a_{17} $[/tex] can now be calculated.
[tex]$ a_{17} = a + (17 - 1)d $[/tex]
[tex]$ \implies a_{17} = 7 + 16(3) $[/tex]
[tex]$ \implies \textbf{a}_{\textbf{17}} \hspace{1mm} \textbf{=} \hspace{1mm} \textbf{55} $[/tex]
Therefore, the [tex]$ 17^{th} $[/tex] term of the sequence is 55.
Hence, the answer.
