... each day is 1 second longer every about 1.5 years
That figure is way off.
According to this Scientific American article, the Earth's rotation rate just after the collision that formed the Moon was about once every 6 hours. At that time, the Moon would have been about 25,000 kilometers away. The tidal effect of the Moon is the major reason the day has been lengthening, and the Moon's orbit has been widening.
The collision is believed to have taken place about 4.5 billion years ago, not long after the formation of the proto-Earth.
There are still some open questions about the impact hypothesis (see the linked Wikipedia article), so this is uncertain.
I strongly suspect that the impact would have erased any information about the Earth's rotation rate before the impact. (It might be possible to estimate the pre-collision rotation rate by modelling the initial formation of the Earth; I don't know whether there's been any research in this area.)
That's not quite correct.
You may have noticed that during summer the days are longer (and the nights shorter) than during winter. That is because the earth's axis is tilted about $23^o$ from the plane of it's orbit around the sun. With this tilt, as the earth travels around the sun the northern hemisphere gets longer days is the north pole is tilted towards the sun. Shorter days happen when it's the reverse. The change happens gradually, with the shortest day at midwinter (approximately 21 December) and the longest day around 21 June. At equinox time (21 March & 21 September) day and night are of equal length.
Once you go north of the Arctic Circle (or south of the Antarctic Circle), which is at $67^o$ north ($67=90-23$), you will find that there are periods of the year when the sun does not rise above the horizon during the day, because the bulge of the earth is in the way. At other times of the year, the sun does not set at all.
However, as in more temperate latitudes, the change is gradual. On 21 December there is indeed no sunlight. Then, as you go into the new year, there comes a day when the sky starts to brighten a bit around noon time. Still later, the sun may come up for a short period. The exact day this happens depends on your latitude (how far north you are). That period gets longer by the day, until on a certain day the sun no longer sets at all, and you have 24 hour daylight. The Arctic Circle is the latitude where there will be a day without sun, and a day without night. Further north more and more days are like that.
Greenland stretches from about $60^o$ to $85^o$ North. Hence the southern end of Greenland always gets at least some sun, even in the middle of winter. ANd, during summer, ther will still be some night.
To add a final bit of confusion: I've been talking all along about the northern hemisphere. If you live in Australia like me, it's the exact reverse. We have winter in June, summer in December.
Best Answer
The length of a day is increasing slowly. The rate is very slow, about 0.0017 seconds per century.
The length of SI day is based upon the mean solar day between 1750 and 1892. A mean day nowadays (!) is about 0.002 seconds longer, which means that we accumulate a difference of about 0.6 seconds every year. This is why we have leap seconds (leap second is when we stop our clocks for one second to let the earth "catch up"). From the Wikipedia article on leap seconds:
Also see: ΔT.