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21. This is the Fibonacci series.
The next number has to be 41; you always add the powers of three to the previous number beginning with 0. 3^0=1; 1+1=2; 3^1=3; 2+3=5; 3^2=9; 5+9=14; 3^3=27; 14+27=41 the next number would be 122 (81+41); you could also multiply the number by 3 and then substract 1. By the way: x(n)=x(n-1)+3^(x-2) ------------------------------------------------------------------------------------------------------------------- It could also be the Catalan number sequence, found by 1/(n+1) * (2n choose n), in which case the next number would be 42.
The next series in the pattern would start at 102 + 666. the pattern is +6, +66, and, presumably, +666. If the pattern were to hold true and continue it should be 768, 769, 770, 771, 772, 773, 774 that is a total of 6 numbers before the next series in the pattern and then 774 + 6666 for 4 numbers... the sequence for the pattern is 2, 5, 3, 6, 4, 7,...
The definition of mode regarding to maths is basically a number that appears most in a series of numbers for example 2,3,6,3,2,3 the mode number would be 3 because there is three 3 two 2 and one 6.