Proof by Induction | Revision Server | Page 1

lost fjord Jan 12, 2025, 1:48 PM

#

help pla

#

I’ve messed up on ii

lost fjord Jan 12, 2025, 2:09 PM

#

@haughty junco

#

<@&791435371564892232>

obtuse pelican Jan 12, 2025, 2:14 PM

#

so base case: f(1) = 81-16=65

Assume true for f(k)
now consider f(k+1)-f(k) = 3^(2k+6) - 2^(2k+2) - 3^(2k+4) + 2^2k
= 3^(2k+4) (3^2 - 1) + 2^2k (1 - 2^2)
= 8 * 3^(2k+4) - 3 * 2^2k
= 5 * 3^(2k+4) + 3(3^(2k+4) - 2^2k)
= 5 * 3^(2k+4) + 3f(k)

#

now as 5 divides f(k)
5 divides 5 * 3^(2k+4) + 3f(k)

#

so 5 divides f(k+1) - f(k)

#

so 5 divides f(k+1)

#Proof by Induction