What in the world
Number Theory
Level
pending
The number (1,000,001)^19 has 115 digits. What is the sum of the 50th, 51st, 52nd, 53rd, 54th, and 55th digits (with the leading digit being the first digit)?
The answer is 29.
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1 solution
Write out the first few terms to notice a pattern-
(1000001)^0 = 1
(1000001)^1 = 10000001
(1000001)^2 = 100000020000001
(1000001)^3 = 1000000300000030000001
The digits follow pascal’s triangle, with the 6-digit number ending at 6n+1 corresponding to the nth column. Since the number actually starts 5 digits before 6n+1, at the (6n-4)th digit, the [56=(n6-4)] n=9th term is what we’re looking for.
On the 19th layer, the 9th term of pascal’s triangle is 19 C 9 = 092378.
0+9+2+3+7+8 = 29.