Question

Every hour a clock chimes as many times as the hour. For example, the clock chimes 4 times at both 4:00 am and 4:00 pm. How many times does it chime from 4:00am through 11:00 pm of the same day including 4:00 am and 11:00 pm.?
What is the proper way of doing this question other than listing all the terms out? Like is there a logical way of thinking about this?
Oh and another math question: If 45 is the sum of n consecutive positive integers, what is th largest possible value of n? I want to know how these 'consecutive numbers' questions work, like how do I list them in my equation?
For the consecutive number stuff, it is always best to recall that

1+2+3+...+n = n(n+1)/2

Then you can multiply that and shift it in various ways.

Or, you can remember your arithmetic series stuff, and recall that the sum of the first n terms of an A.P. is

n/2 (T1+Tn) = n/2 (2a + (n-1)d)

So, for the clock, you have

4+5+...+12 + 1+2+...+11
= (1+2+...+12)-(1+2+3) + (1+2+...+11)
= 12*13/2 - 2*3/2 + 11*12/2
= 78-6+66
= 138
Ok thank you! What about the second question I posted?

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