Question

Prove that for any the expression 3^(2n+2) - 9 is divisible by 72. Do not use Mathematical Induction.4

Answer 1

Answer:

Let p(x)=32n+2−8x−9 is divisible by 64                      …..(1)

When put n=1,

p(1)=34−8−9=64 which is divisible by 64

Let n=k and we get

p(k)=32k+2−8k−9 is divisible by 64

32k+2−8k−9=64m      where m∈N                      …..(2)

Now we shall prove  that p(k+1) is also true

p(k+1)=32(k+1)+2−8(k+1)−9 is divisible by 64.

Now,

p(k+1)=32(k+1)+2−8(k+1)−9=32.32k+2−8k−17

=9.32k+2−8k−17

=9(64m+8k+9)−8k−17

=9.64m+72k+81−8k−17

=9.64m+64k+64

=64(9m+k+1

Answer 2

Answer:

Answer:

Let p(x)=32n+2−8x−9 is divisible by 64                      …..(1)

When put n=1,

p(1)=34−8−9=64 which is divisible by 64

Let n=k and we get

p(k)=32k+2−8k−9 is divisible by 64

32k+2−8k−9=64m      where m∈N                      …..(2)

Now we shall prove  that p(k+1) is also true

p(k+1)=32(k+1)+2−8(k+1)−9 is divisible by 64.

Now,

p(k+1)=32(k+1)+2−8(k+1)−9=32.32k+2−8k−17

=9.32k+2−8k−17

=9(64m+8k+9)−8k−17

=9.64m+72k+81−8k−17

=9.64m+64k+64

=64(9m+k+1