Using induction to prove:

for n is positive integer that 8^n - 14n = 27 is divisible by 7

Question

Using induction to prove:

 for n is positive integer  that 8^n - 14n = 27 is divisible by 7

anonymous · Answer

so for my first step I have proven that this DOES hold true from n=1 

now assume this holds true from some value of n=k.
 so 8^k-14k=27 is divisible by 7

show that for k+1 = 8^(k+1)- 14(k+1) +27 is divisible by 7

so 8^(k+1)= 8*8^k - 14k +27-14
i was thinking of breaking this into 8*8^k= (7+1)8^k
since I know 
8^k -14k+27 is divisible by 7 and 7(whatever is left) is divisible by seven.... it must be divisible by seven..... 

but when i try to figure it out the numbers dont come out right....

zarkon · Answer

$$8^{k+1}-14(k+1)+27$$

$$8\cdot8^{k}-14k-14+27$$

$$8\cdot8^{k}-(8-7)14k+(8-7)27-14$$
$$8\cdot8^{k}-8\cdot 14k+8\cdot 27+7\cdot 14k-7\cdot 27-14$$

$$8[8^k-14k+27]+7\cdot 14k-7\cdot 27-14$$

$$8[8^k-14k+27]+7[ 14k-27-2]$$

zarkon · Answer

even better...nice work