prove this by induction (1 Viewer)

[phoenix159]

Prove by induction: 5ⁿ + 6ⁿ - 1 is divisible by 10 for all integers n

 

[integral95]

for n = 1 LHS = 10

Assume true for n = k where k is an integer i.e

for n = k+1 (I'll be skipping most of the algebra)



now 5 - 5^k is divisible by 10 since 5^k will always have the last digit 5 and so there fore 5 - 5^k has the last digit of 0 hence the whole expression is divisible by 10 so there fore it's true for n = k+1

Edit: btw this is only true for all positive n integers NOT all integers

 

[phoenix159]

Thanks!

 

[RealiseNothing]

All that is left is to prove is odd since always ends in a 5. This is obvious as all the powers of 6 are even, then the on the end makes it odd. #