Question

The first two terms of a sequence are 5 and 7. Each term after the second is found by taking the average (arithmetic mean) of all the preceding terms. What is the 50th term in this sequence

Answer 1

Answer:

6

Step-by-step explanation:

We can prove that every number after the second will be a six by induction.

Base case. Since $\frac{5 + 7}{2} = 6$, so the third term is a six.

Inductive hypothesis. Fix the number of terms to be n and assume that

$\frac{1}{n} \sum\limits_{i=1}^{n}t_i = 6$

Inductive step. We will now show that $\frac{1}{n+1} \sum\limits_{i=1}^{n+1}t_i = 6$.

Notice that

$\begin{array}{lll}\frac{1}{n+1} \sum\limits_{i=1}^{n+1}t_i & = \frac{n}{n(n+1)} \sum\limits_{i=1}^{n+1}t_i & \\& = \frac{t_{n+1}}{n+1} + \frac{n}{n(n+1)} \sum\limits_{i=1}^{n}t_i &\\& = \frac{t_{n+1}}{n+1} + \frac{6n}{n+1} & \text{(by the IH)}\\& = \frac{6}{n+1} + \frac{6n}{n+1} & \text{by de\\finition}\\& = \frac{6(n+1)}{n+1} & \\& = 6 & \end{array} \square$