Prove inequality consisting of sum using mathematical induction

Question

I wish to seek help for a question from the Finnish matriculation exam in mathematics. There are multiple methods to solve the exercise, but I am ONLY interested in solving it using mathematical induction. The question is as follows:

$$\text{let}\ n\ \text{be a positive integer}, n\ge1$$ $$\text{Show that:}$$ $$\frac{127}{7}\cdot(n-1)^7\le\sum_{k=n}^{2n-1}k^6\le\frac{127}{7}\cdot\ n^7$$

I have tried hard to solve this using induction, but I cannot get very far at all. Any help would be appreciated.

EDIT: What I have done so far.

1. Test with n = 1, which gives:

$$0\le1\le\frac{127}{7}, \text{which is true}$$

2. Assumption:

$$\frac{127}{7}\cdot(n-1)^7\le\sum_{k=n}^{2n-1}k^6\le\frac{127}{7}\cdot\ n^7$$

3. Induction step:

$$\frac{127}{7}\cdot\ n^7\le\sum_{k=n+1}^{2n+1}k^6\le\frac{127}{7}\cdot\ (n+1)^7$$

But I don't know how to prove this in terms of the assumption. I would assume that there is some way to remove the sum from this inequality. It seems to be possible to expand it using my CAS-calculator. But I just can't wrap my head around it.

I hope this enough explanation of what I have done so far.

(I might have used the wrong mathematical terms here, since English is not my first language)

Answer 1

Thanks to the help of @maxmilgram, I've managed to figure out the solution.

SOLUTION:

1. Test with n = 1, which gives:

$$0\le1\le\frac{127}{7}, \text{which is true}$$

2. Assumption:

$$\frac{127}{7}\cdot(n-1)^7\le\sum_{k=n}^{2n-1}k^6\le\frac{127}{7}\cdot\ n^7$$

3. Induction step:

$$\frac{127}{7}\cdot\ n^7\le\sum_{k=n+1}^{2n+1}k^6\le\frac{127}{7}\cdot\ (n+1)^7$$

And since:

$$\sum_{k=n+1}^{2n+1}k^6=\sum_{k=n}^{2n-1}k^6-n^6+(2n)^6+(2n+1)^6$$

we can write the induction step in terms of the assumption:

$$\frac{127}{7}n^7\le\sum_{k=n}^{2n-1}k^6-n^6+\left(2n\right)^6+\left(2n+1\right)^6\le\frac{127}{7}\left(n+7\right)^7$$

$$\frac{127}{7}n^7+n^6-\left(2n\right)^6-\left(2n+1\right)^6\le \sum _{k=n}^{2n-1}k^6\le \frac{127}{7}\left(n+7\right)^7+n^6-\left(2n\right)^6-\left(2n+1\right)^6$$

In other words, if we could prove that:

$$\frac{127}{7}n^7+n^6-\left(2n\right)^6-\left(2n+1\right)^6\le \frac{127}{7}\left(n-1\right)^7$$ and

$$\frac{127}{7}\left(n+7\right)^7+n^6-\left(2n\right)^6-\left(2n+1\right)^6\ge \frac{127}{7}n^7$$

we would know that the induction step is indeed true. After some simplification we end up with:

$$-573n^5+395n^4-795n^3+321n^2-139n+\frac{120}{7}\le 0, \text{which is true since } n\ge 1$$

$$189n^5+395n^4+475n^3+321n^2+115n+\frac{120}{7}\ge 0, \text{which is true since } n \ge 1$$

In other words, we have shown that the induction step is true if the induction assumption also is true. Therefore, the statement is true for all positive integers n.