Because Earth's rotation is gradually slowing, the length of each day increases: The day at the end of 1.0 century is 1.0 ms longer than the day at the start of the century. In 82 centuries, what is the total of the daily increases in time (that is, the sum of the gain on the first day, the gain on the second day, etc.) ?

Step-by-step explanation: Ok, you know that the day roughly grows by 1.0ms for century.

now i understand you only want to know the daily increases over the 82 centuries ...

a centuri have 100 years, a year has 365 days, so te century has 3650 days

then the first century has no displacement in time, is the current one.

the second has 100*365*0.1 ms

the tird has 100*365*02 ms

and goes on, the total displacement is 100*365 * (0.1 + 0.2 + ... + 0.81)

wich is a series of the form (1/10) * ∑n between 1 and 81, wich solution is (1/10) * 0.5*81 * (81 + 1)

so 3650*0.3321 = 1212.165 ms will be the time increased over 82 centuries ... or 1.2 seconds.