Question

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.

Answer 1

We will use mathematical induction to prove this inequality.

Step-by-step explanation:

We will prove this inequality using mathematical induction. First, let's check the base case when n = 0. The left-hand side (LHS) of the inequality is 0 and the right-hand side (RHS) is (4*0)^4 = 0. So, the inequality holds for n = 0.

Next, assume that the inequality holds for some positive integer k, i.e.,

∑([2i]^4) ≤ (4k)^4 (where the sum is taken from i = 0 to k)

We will prove that it also holds for k + 1. Adding the (k+1)th term to both sides of the inequality:

∑([2i]^4) + ([2(k+1)]^4) ≤ (4k)^4 + ([2(k+1)]^4)

Now, simplifying the LHS and RHS:

(∑([2i]^4)) + ([2(k+1)]^4) ≤ (4k)^4 + ([2(k+1)]^4)

Answer 2

The statement is true for every n between 0 and 77 and it is false for
`n\geq 78`

Step-by-step explanation:

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

For n=0:
`\sum^(n)_(i=0) (2i)^4=0 \leq 0=(4n)^4`

For n=1:
`\sum^(n)_(i=0) (2i)^4=16 \leq 256=(4n)^4`

From this point we will assume that
`n\geq 2`

As we can see,
`\sum^(n)_(i=0) (2i)^4=\sum^(n)_(i=0) 16i^4=16\sum^(n)_(i=0) i^4` and
`(4n)^4=256n^4`. Then,

`\sum^(n)_(i=0) (2i)^4 \leq(4n)^4 \iff \sum^(n)_(i=0) i^4 \leq 16n^4`

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

`\sum^(n)_(i=0) i^4=(n^5)/(5) +(n^4)/(2) +(n^3)/(3)-(n)/(30)=(6n^5+15n^4+10n^3-n)/(30)`

Therefore:

`\sum^(n)_(i=0) i^4 \leq 16n^4 \iff (6n^5+15n^4+10n^3-n)/(30) \leq 16n^4 \\\\ \iff 6n^5+10n^3-n \leq 465n^4 \iff 465n^4-6n^5-10n^3+n\geq 0`

and, because
`n \geq 0`,

`465n^4-6n^5-10n^3+n\geq 0 \iff n(465n^3-6n^4-10n^2+1)\geq 0 \\\iff 465n^3-6n^4-10n^2+1\geq 0 \iff 465n^3-6n^4-10n^2\geq -1\\\iff n^2(465n-6n^2-10)\geq -1`

Observe that, because
`n \geq 2` and is an integer,

`n^2(465n-6n^2-10)\geq -1 \iff 465n-6n^2-10 \geq 0 \iff n(465-6n) \geq 10\\\iff 465-6n \geq 0 \iff n \leq (465)/(6)=(155)/(2)=77.5`

In concusion, the statement is true if and only if n is a non negative integer such that
`n\leq 77`

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

Note: If you compute
`(4n)^4- \sum^(n)_(i=0) (2i)^4` for 77 and 78 you will obtain:

`(4n)^4- \sum^(n)_(i=0) (2i)^4=53810064`

`(4n)^4- \sum^(n)_(i=0) (2i)^4=-61754992`