Show that 2^2n-1 +1 is divisible by 3 for all n > 1.

Step-by-step explanation:

Let's assume that

is divisible by 3.

for n= 1

       = 3

Hence, P(1) is true for n=1

for n=2

       = 9, which is divisible by 3.

As we can see, P(n) is also true for n= 2.

Let's say P(n) is true for n = k

So,

is divisible by 3.

For P(n) should be true, the difference of P(k+1) and P(k) must will have divisible by 3.

So,

as P(k+1)-P(k) is divisible by 3.

As a result, P(n) is true for all n>1.

walter needs to make at least 5 more cookies but no more than 40. 5 ≤ x ≤ 40

step-by-step explanation:

20 ≤ x + 15 ≤ 55

subtract 15 from all parts

20-15 ≤ x + 15-15 ≤ 55-15

5 ≤ x ≤ 40

this means how many more cookies he has to make

walter needs to make at least 5 more cookies, but no more than 40 cookies; 5 ≤ x ≤ 40