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Question: A sustainable infrastructure project requires materials costing $ 3x + 5y $ dollars and $ 2x - y $ dollars. If the total cost is $ 20 $ dollars and $ x = 2 $, find $ y $.
Solution: The sequence is arithmetic with first term $ a_1 = 2 $, common difference $ d = 3 $, and last term $ a_n = 50 $. Using the formula for the $ n $-th term of an arithmetic sequence:
Question: A community solar project generates 150 kWh in its first month, 180 kWh in the second, and increases by 30 kWh each month. How many months until the total energy produced reaches 1200 kWh?
Solution: The total energy after $ n $ months is the sum of the arithmetic series:
Here, $ a_1 = 150 $, $ d = 30 $, and $ S_n = 1200 $. Substituting:
$$Question: A linguist observes that the number of unique words in a text grows with the square of the page count. If a 10-page document contains 220 unique words, how many unique words would a 15-page document have, assuming the growth pattern holds?
Solution: Substitute $ x = 2 $ and $ y = -1 $ into the expression for $ E $:
So $ t > 13.51 $, meaning the population drops below 500 in year $ 2020 + 14 = 2034 $.
Solution: Start by recalling the identity:
Since $ t = 1.5 $ corresponds to 1.5 years after 2020, the minimum occurs during the year 2021 (halfway through). However, since the question asks for the year, we report the integer year closest to the minimum: