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.

1 Answer
George C.
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"#

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