Work problem…piano practice

Question

Yvonne's piano teacher asked Yvonne to practice the piano for five minutes the first night, 10 minutes the second night, 15 minutes the third night, 20 minutes the fourth night and she doesn't have to practice on Sundays. How many nights will it take your von to have three hours of practice?

Answer 1

Consider the sequence of minutes for consecutive practices: 5, 10, 15, 20,... can be rewritten as (5·1), (5·2), (5·3), ... . When we sum up this sequence, we then have 5·(1 + 2 + 3 + 4 + ...) or 5 times a sum of consecutive positive integers. We can then employ the idea that the sum of consecutive integers from 1 to some integer n is n(n + 1)/2. So our series can be written as 5n(n + 1)/2 for some number n.

 

Now we. note that 3 hours is 180 minutes. So our set-up is then 5n(n + 1)/2 =180, or n^2 + n =72. Subtracting 72 from both sides sets up a factorable quadratic, n^2 + n - 72 = 0 or (n+9)(n-8) = 0; this gives us n = -9 or 8. Since we started that n is a POSITIVE integer, -9 doesn't make sense, so n = 8. 

 

Note that 1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 = 36, and 5 · 36 = 180 minutes or 3 hours of practice. 

 

But, she takes Sunday off, and no matter what day of the week she starts she'll hit  a Sunday... so in 9 days from the first day of practice, she'll achieve that 3 hour target time.