Prove or disprove (from i=0 to n) sum([2i]^4) < = (4n)^4. if true use induction, else give the smallest value of n that it doesn't work for.

The statement is true for every n between 0 and 77 and it is false for

Step-by-step explanation:

First, observe that, for n=0 and n=1 the statement is true:

For n=0:

For n=1:

From this point we will assume that

As we can see, and . Then,

Now, we will use the formula for the sum of the first 4th powers:

Therefore:

and, because ,

Observe that, because and is an integer,

In concusion, the statement is true if and only if n is a non negative integer such that

So, 78 is the smallest value of n that does not satisfy the inequality.

Note: If you compute   for 77 and 78 you will obtain:

a

b

d

step-by-step explanation:

the answer is y < -3 or   y > 3

-3y > 7 + 2

-3 y > 9

y < -3