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2 problems match “linear recurrences”. Clear

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1. First term of a recurrence with a constant subtraction ●●○○○ 20 ptsA sequence $a_1, a_2, a_3, \ldots$ satisfies $a_n = 3a_{n-1} - x$ for every $n \ge 2$, where $x$ is an integer that is the same for all terms. Given that $a_2 = 7$ and $a_4 = 43$, find $a_1$.
2. Dragon burning a village ●●●○○ 50 ptsTrodgor the dragon is burning down a village consisting of 90 cottages. At time $t=0$ an angry peasant arises from each cottage, and every 8 minutes (480 seconds) thereafter another angry peasant spontaneously generates from each non-burned cottage. It takes Trodgor 5 seconds to burn a peasant or to burn a cottage, but Trodgor cannot begin burning cottages until all the peasants around him have been burned. How many seconds does it take Trodgor to burn down the entire village?