How can we prove that, each term in a sequence 12,1122,111222........ is the product of two consecutive numbers?
the sequence goes like this 3 * 4, 33 * 34, 333 * 334 ............ The question is of 7 marks, so, I guess we can't use hit and try method.
the sequence goes like this 3 * 4, 33 * 34, 333 * 334 ............ The question is of 7 marks, so, I guess we can't use hit and try method.
1 Answer
Jun 12, 2018
See explanation...
Explanation:
Note that:
#3 = 1/3(10^1 - 1)#
#33 = 1/3(10^2 - 1)#
#333 = 1/3(10^3 - 1)#
In general:
#overbrace(33...33)^"n digits" = 1/3(10^n - 1)#
So:
#overbrace(33...33)^"n digits" * overbrace(33...3)^"n-1"4=1/3(10^n-1) * 1/3(10^n+2)#
#color(white)(33...33 * 33...34) = 1/9(10^(2n)+10^n-2)#
#color(white)(33...33 * 33...34) = 1/9(10^(2n)-1)+1/9(10^n-1)#
#color(white)(33...33 * 33...34) = overbrace(11...11)^"2n digits"+overbrace(11...11)^"n digits"#
#color(white)(33...33 * 33...34) = overbrace(11...11)^"n digits"overbrace(22...22)^"n digits"#