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

6

Explanation:

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

Base case. Since
`(5 + 7)/(2) = 6`, so the third term is a six.

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

`(1)/(n) \sum\limits_(i=1)^(n)t_i = 6`

Inductive step. We will now show that
`(1)/(n+1) \sum\limits_(i=1)^(n+1)t_i = 6`.

Notice that

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