Answer:
6
Step-by-step explanation:
We can prove that every number after the second will be a six by induction.
Base case. Since [tex]\frac{5 + 7}{2} = 6[/tex], so the third term is a six.
Inductive hypothesis. Fix the number of terms to be n and assume that
[tex]\frac{1}{n} \sum\limits_{i=1}^{n}t_i = 6[/tex]
Inductive step. We will now show that [tex]\frac{1}{n+1} \sum\limits_{i=1}^{n+1}t_i = 6[/tex].
Notice that
[tex]$\begin{array}{lll}\frac{1}{n+1} \sum\limits_{i=1}^{n+1}t_i & = \frac{n}{n(n+1)} \sum\limits_{i=1}^{n+1}t_i & \\& = \frac{t_{n+1}}{n+1} + \frac{n}{n(n+1)} \sum\limits_{i=1}^{n}t_i &\\& = \frac{t_{n+1}}{n+1} + \frac{6n}{n+1} & \text{(by the IH)}\\& = \frac{6}{n+1} + \frac{6n}{n+1} & \text{by de\\finition}\\& = \frac{6(n+1)}{n+1} & \\& = 6 & \end{array} \square$[/tex]
