Question

What is the 1000th digit to the right of the decimal point in the decimal representation of (1 + sqrt 2)3000 ?

Answer 1

hint: Expand (1 − root 2)n using the binomial theorem.

Answer 2

*hint: Expand (1 − root 2)^n using the binomial theorem

Answer 3

The 1000th digit to the right of the decimal point in the decimal representation of (1 + root 2)^3000 is 9.

Answer 4

9
From Mathematica:
N[(1+ Sqrt[2])3000,1004] <- "list 1004 digits for the answer"
7242.6406871192851464050661726290942357090156261308442195300392139721974353863211165511626029829247182050415386927368910747745081675522116379323644912998074942396679977825167782673998515034583461817144103286799148179108236037905860441855522559226675958286569876914529261429643525192878108839857542239690617601896389954264889218618457750571851642372508632798851895067256610125955710340631238102254965414811606999120970144450916320833709493627434692054788107558647417353893347900061390390468557069617117058655277945837484931462655002612857058180440474162314307564622466971209032961160679456341047576404862275424767971678006256753399564375196681833112779775400934164739404471247890301784576112009559441026584691834871923135217910643941219565768351966120735273683100680823382720881841487214326902512055109605105352177398094187182288990824141426964471186991540980407423261887807374577156543013379526451773360748324025185594434317410810113231927548607944130513335051749745180668259276149746456384688555465416269

Answer 5

Thank you for the medal.

Answer 6

Consider a(n) = (1+root 2)^n + (1-root 2)^n.

Expanding both terms using the binomial theorem, notice that the odd powers cancel, while the coefficients of even powers are all integers, and therefore An is an integer.

Then, |1 − root 2| < 1, and so (1 − root 2)^n tends to zero as n tends to infinity.
Using logarithms and/or a calculator, we find that 10^−1149 < (1 − root 2)^3000 < 10^−1148.

Therefore (1 + root 2)^3000 has 1148 nines to the right of the decimal point, and so the 1000th such digit is a 9.

Answer 7

np robtobey :)

Answer 8

See http://oeis.org/A002193/b002193.txt

search by 1001 element.