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There are $52$ weeks to expiration. The price at the end only depends on how many up weeks there are, not on the order of the ups and downs. If there are $26$ up weeks and $26$ down weeks, the stock price will be $25(1.05)^{26}(.95)^{26}\approx 23.42$ so the put will be worth $1.58$. What is the value of $1.58$ a year from now?
The sum will vary over the number of up weeks and down weeks. You don't say, but we are probably expected to assume that up and down weeks are equally probable. The chance of $26$ up weeks is then ${52 \choose 26} 2^{-52}$ You need to add up the values of the put times their probability over the number of up weeks. Note that if there are too many up weeks the value of the put is zero and you can ignore any more ups than that in your sum.
we can use a binomial tree with risk-neutral pricing to model this, assuming a continuous dividend rate $\delta$ and letting the risk free interest rate be $r$. Although the stock does not pay dividends, we can still use this general formula.
Assuming 52 weeks in a year, we will let $$u= 1.05, d= 0.95, p^* = \frac{e^{(r-\delta)/52}-d}{u-d}$$ where $p^*$ is the risk-neutral probability of an up move.
Then the price of a put is equal to
$$P=e^{-r}\sum_{i=0}^{52}\binom{52}{i}(p^*)^i(1-p^*)^{52-i}\max(0, 25 - 25u^id^{52-i})$$