What is the units digit of the 1200th number in the Fibonacci sequence?

Question

lol, I can only think of doing it the really long way...

Answer 1

If you write the Fibonacci sequence but only take the last digit (mod 10), you'll find that it repeats after 60 digits.

Fibonacci mod 10:
1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5,
6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5,
1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0,

1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5,
6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5,
1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0,

1, 1, 2, 3, 5, 8, 3, 1, 4, 5, 9, 4, 3, 7, 0, 7, 7, 4, 1, 5,
6, 1, 7, 8, 5, 3, 8, 1, 9, 0, 9, 9, 8, 7, 5, 2, 7, 9, 6, 5,
1, 6, 7, 3, 0, 3, 3, 6, 9, 5, 4, 9, 3, 2, 5, 7, 2, 9, 1, 0,
etc.

So the units digit of the 1200th number will be the same as the units digit of the 60th number.

As you can see above, the 60th number ends with 0.

P.S. Some people start the Fibonacci sequence with a "zeroth" term being 0, then follow 1, 1, 2, 3, 5, etc. If you look at the sequence this way, you don't even have to go to 60 terms because it is the same as the "zeroth" term. Either way the answer is 0.

In general, if someone asked you for the units digit of the "n"-th term, you would just subtract the next lowest multiple of 60. So if someone asked for the units digit of the 187th term, you would subtract 180 (same as taking mod 60). Thus the digit would be the same as the 7th term.

Answer 2

You can be quicker than that.

F(15) is 610, units digit zero.

F(30) = F(15) * (F(16) + F(14)) so without even looking at F16 or F14, we know that F(30) has a units digit of zero, too.

F(45) = F(30) * F(16) + F(15) * F(29), so again without knowing F(29) or F(16), we know that the units digit of F(45) is zero. The formula is one of the very many "recurrence relationships" for Fibonacci numbers, and you can look them up on Wikipedia, or in Mathworld Wolfram.

And at every multiple of 15, the Fibonacci number is made up of multiples of Fibonacci numbers which have occurred at smaller multiples of 15, and therefore they all have the units digit equal to zero, too.

Answer 3

Puzzling is correct. But it can be taken 1 step further to make it easier.

60th term would be the same as the 0th term. This term exists. The beginning of the sequence - 1, 1, 2, 3, 5, 8 - are considered terms 1 to 6. Sometimes the sequence will start 0, 1, 1, 2, 3, 5, 8... The 0 is considered term 0.

So term 0, 60, 120... 1140, 1200, all have a units digit of 0.

EDIT: Puzzler's P.S. just covered what I said.

Answer 4

According to

http://en.wikipedia.org/wiki/Fibonacci_number

There is a closed form solution for the Fibonacci sequence so that you could calculate the 1200th number without doing it the long way.

However the number may be so large that you will not be able to find the units digit.

Answer 5

The number would be humongous! Write down the first 20 or so numbers and see if there is a last digit sequence that repeats.

Answer 6

Use a calculator to find the first 80 or so figures. The ones digits begin to repeat around n = 70 or so.

Answer 7

Since Fa divides Fab for all natural a,b and 15 divides 1200

we have F15 divides F1200 but F15 = 610. Therefore

F1200 ends in zero.

Answer 8

The 1200th number in this series is 2.7 X 10^250.