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?

Well, let's put on our thinking caps! Each hour, the clock chimes as many times as the hour itself. So at 4:00 am, it chimes 4 times. At 5:00 am, it chimes 5 times, and so on.

Now, if we consider the AM hours from 4:00 am to 11:00 am, we have a total of 8 hours. So, if we add up the number of chimes for each hour, we get:
4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 = 60 chimes

In the PM hours from 12:00 pm to 11:00 pm, we have another 12 hours. However, in the PM, we have to subtract 12 from the hour to sync it with our 12-hour clock. So, for example, at 4:00 pm, we have to substract 12 from 16, giving us 4 chimes. Therefore, the PM hours would be:
1 + 2 + 3 + 4 + 5 + 6 + 7 + 8 + 9 + 10 + 11 + 12 = 78 chimes

Adding together the chimes from the AM and PM hours, we get a grand total of 60 + 78 = 138 chimes.

Phew! So, the clock chimes 138 times from 4:00 am through 11:00 pm. I hope I didn't chime in too much with that explanation!

Yes, there is a logical way to approach this question without listing out all the terms.

First, let's understand the pattern of the clock chimes. The clock chimes the same number of times as the hour. So at 4:00 am and 4:00 pm, it chimes 4 times. At 5:00 am and 5:00 pm, it chimes 5 times, and so on.

To find the total number of chimes from 4:00 am through 11:00 pm, we can break it down into three parts:

1. From 4:00 am to 3:59 pm: This covers the chimes from 4:00 am through 11:59 am, as well as the chimes from 12:00 pm through 3:59 pm. The chimes during this period can be calculated using the sum of arithmetic series formula:
Number of terms = (Last term - First term) / Common difference + 1
In this case, the first term is 4, the last term is 12, and the common difference is 1 (since it increases by 1 each hour).
So, the number of chimes during this period is (12 - 4) / 1 + 1 = 9 + 1 = 10.

2. From 4:00 pm to 11:00 pm: This covers the chimes from 4:00 pm through 10:59 pm. Using the same formula as above, the number of chimes during this period is:
(10 - 4) / 1 + 1 = 6 + 1 = 7.

3. At 11:00 pm: At 11:00 pm, the clock chimes 11 times.

To find the total number of chimes, we add up the chimes from each part: 10 + 7 + 11 = 28.

Therefore, the clock will chime a total of 28 times from 4:00 am through 11:00 pm of the same day.