An arts academy requires there to be 5 teachers for every 90 students and 3 tutors for every 30 students. How many students does
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Answers:
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Explanation:
Note we subtract 3 off the previous term (t1) to get the next term (t2). Each new successive term is found this way
and so on. This process may take a while to reach
There's a shortcut. The nth term of any arithmetic sequence is
We plug in and simplify
Then we can plug in various positive whole numbers for n to find the corresponding value. For example, plug in n = 2
which matches with the second term we found earlier. And,
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The notation refers to the sum of the first ten terms
We could use either the long way or the shortcut above to find all through
. Then add those values up. Or we can take this shortcut below.
The sum of the first ten terms is -85
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As a check for , here are the first ten terms:
- t1 = 5
- t2 = 2
- t3 = -1
- t4 = -4
- t5 = -7
- t6 = -10
- t7 = -13
- t8 = -16
- t9 = -19
- t10 = -22
Then adding said terms gets us...
5 + 2 + (-1) + (-4) + (-7) + (-10) + (-13) + (-16) + (-19) + (-22) = -85
This confirms that is correct.