Camsie is starting a new job where she earns n dollars on her nth day of work. What is the total amount that Camsie will earn for the first ten days of work at her new job?
On each of the first 10 days of work at her new job, Camsie will earn 1, 2, 3, 4, 5, 6, 7, 8, 9 and 10 dollars, respectively. That’s a total of 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 = 10(1 + 10)/2 = 5(11) = 55 dollars.
How many days will Camsie have worked at her new job when her cumulative earnings total 595 dollars?
Camsie’s cumulative earnings for working the first n days at her new job equal the sum of the integers from 1 to n. Since 1 + 2 + 3 + … + n = (n/2)(n + 1), we can solve the equation (n/2)(n + 1) = 595 → n(n + 1) = 1190 → n2 + n − 1190 = 0. Factoring the quadratic yields (n − 34)(n + 35) = 0. So, n − 34 = 0 → n = 34 and n + 35 = 0 → n = −35. Therefore, Camsie’s cumulative earnings will total 595 dollars when she has worked 34 days.
What is the minimum number of days Camsie must work at her new job for her cumulative earnings to exceed 1000 dollars?
Using the same expression for the sum of the integers from 1 to n, we have the following inequality: (n/2)(n + 1) > 1000 → n(n + 1) > 2000. Simplifying yields the quadratic expression n2 + n − 2000, which cannot be factored. However, we are essentially looking for two consecutive integers, n and n + 1, that have a product greater than 2000. Since 402 = 1600 and 502 = 2500, we know the integers are between 40 and 50. The square of 45 is 2025, so we check 44 × 45 and 45 × 46 and get 1980 and 2070, respectively. Therefore, we conclude that for Camsie’s cumulative earnings to exceed 1000 dollars, she must work at least 45 days.
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