10,10,12,12,13,13... Sequence formula

[raztactical]

What is the formula for the following sequence 10,10,11,11,12,12,13,13,14,14...
How many n terms would the sequence contain before reaching 2000 or more.

I'm trying to figure out how long it will take me to learn 2000 kanji if i start at 10 per day and add 1 extra every 2 days.

 

[Ishuda]

Approaching the problem a slightly different but equivalent way (to possibly make it easier to solve but to not actually do it that way), assume we break this into two identical patterns
10, 11, 12, 13, ..., n
in which we do 1000 twice. Then, if 'by adding one extra every 2 days', you mean learn 10 on each of the first two days, then 11 on days three and four, 12 on days five and six, etc. then the answer can be derived using Denis's method (realizing, of course, that the n is actually half of the number of days).

However, if you meant learn one more every two days, then the answer is quite a bit larger (~5 and a half years)

 

[raztactical]

Thanks a lot

I can't believe i didn't think of approaching the problem that way.
So it would come up with 73 days and i would have to learn 45 new Kanji on last 2 days +20 on the vary last day.

Since:
990 = [n(n + 1) / 2] - 45 = 36n
990*2=1980 (+20 on last day)
36*2=72
+1=73

Is this correct?

 

[raztactical]

CORRECT!

And since 46 would be received on 73rd day, but only 20 is needed,
and assuming the 46 is received as 1 every 1/46th of the day,
then the actual total time is 72 days, 10 hours, 26 minutes and 05 seconds

So keep an eye on your watch: 10:26:05 am on 73rd day

Yeah i don't think i need to be that precise plus there's loads more than just 2000 but i set that as my target for now.

Thanks a lot for your help guys.