QuestionAugust 25, 2025

Consider a population that grows according to the recursive rule P_(n)=P_(n-1)+70 with initial population P_(0)=60. Then P_(1)=square P_(2)=square Find an explicit formula for the population. Your formula should involve ven (use lowercasen) P_(n)=square Use your explicit formula to find P_(100) P_(100)=square

Consider a population that grows according to the recursive rule P_(n)=P_(n-1)+70 with initial population P_(0)=60. Then P_(1)=square P_(2)=square Find an explicit formula for the population. Your formula should involve ven (use lowercasen) P_(n)=square Use your explicit formula to find P_(100) P_(100)=square
Consider a population that grows according to the recursive rule P_(n)=P_(n-1)+70 with initial population P_(0)=60. Then P_(1)=square P_(2)=square Find an explicit formula for the population. Your formula should involve ven (use lowercasen) P_(n)=square Use your explicit formula to find P_(100) P_(100)=square

Solution
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Answer

P_1 = 130 ### P_2 = 200 ### P_n = 60 + 70n ### P_{100} = 7060 Explanation 1. Calculate P_1 Use the recursive rule P_{n} = P_{n-1} + 70 with P_0 = 60: P_1 = 60 + 70 = 130. 2. Calculate P_2 Apply the recursive rule again: P_2 = P_1 + 70 = 130 + 70 = 200. 3. Derive explicit formula for P_n Recognize the pattern: P_n = P_0 + 70n. Substitute P_0 = 60: P_n = 60 + 70n. 4. Calculate P_{100} using explicit formula Substitute n = 100 into P_n = 60 + 70n: P_{100} = 60 + 70 \times 100 = 7060.

Explanation

1. Calculate $P_1$<br /> Use the recursive rule $P_{n} = P_{n-1} + 70$ with $P_0 = 60$: $P_1 = 60 + 70 = 130$.<br /><br />2. Calculate $P_2$<br /> Apply the recursive rule again: $P_2 = P_1 + 70 = 130 + 70 = 200$.<br /><br />3. Derive explicit formula for $P_n$<br /> Recognize the pattern: $P_n = P_0 + 70n$. Substitute $P_0 = 60$: $P_n = 60 + 70n$.<br /><br />4. Calculate $P_{100}$ using explicit formula<br /> Substitute $n = 100$ into $P_n = 60 + 70n$: $P_{100} = 60 + 70 \times 100 = 7060$.
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