Finding Tha maxima and Minima of a Function given a constraint

If $a_i = 1 - \frac{1}{N_i}$ and $\sum\limits_{I=0}^k{N_i} = n$ , then what is the maximum and minimum values of $\sum\limits_{I=0}^k{a_i}$ ?

Please help, I've tried to solve it but then I got confused. I think I may of found the minimum value to be $\frac{n - k}{n - k + 1}$ but I'm not sure.

Also $N_i > 0$ and $N_i$ is a subset of $\mathbb{N}$ .

#Algebra

Easy Math Editor

This discussion board is a place to discuss our Daily Challenges and the math and science related to those challenges. Explanations are more than just a solution — they should explain the steps and thinking strategies that you used to obtain the solution. Comments should further the discussion of math and science.

When posting on Brilliant:

Use the emojis to react to an explanation, whether you're congratulating a job well done , or just really confused .
Ask specific questions about the challenge or the steps in somebody's explanation. Well-posed questions can add a lot to the discussion, but posting "I don't understand!" doesn't help anyone.
Try to contribute something new to the discussion, whether it is an extension, generalization or other idea related to the challenge.
Stay on topic — we're all here to learn more about math and science, not to hear about your favorite get-rich-quick scheme or current world events.

Markdown	Appears as
`italics` or `_italics_`	italics
`bold` or `__bold__`	bold
- bulleted - list	bulleted list
1. numbered 2. list	numbered list
Note: you must add a full line of space before and after lists for them to show up correctly
paragraph 1 paragraph 2	paragraph 1 paragraph 2
`[example link](https://brilliant.org)`	example link
`> This is a quote`	This is a quote
# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"	# I indented these lines # 4 spaces, and now they show # up as a code block. print "hello world"

Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

$\displaystyle \sum_{i=0}^{k}a_{i} = k - \displaystyle \sum_{i=0}^{k}\dfrac{1}{N_{i}}$
Using AM-GM-HM inequality,
$\dfrac{\displaystyle \sum_{i=0}^{k}N_{i}}{k} \ge \dfrac{k}{\displaystyle \sum_{i=0}^{k}\dfrac{1}{N_{i}}}$
$\displaystyle \sum_{i=0}^{k} \dfrac{1}{N_{i}} \ge \dfrac{k^{2}}{n}$
$-\displaystyle \sum_{i=0}^{k} \dfrac{1}{N_{i}} \le - \dfrac{k^{2}}{n}$
$\displaystyle \sum_{i=0}^{k} a_{i} \le k - \dfrac{k^{2}}{n}$
This is the maximum value of the expression, I am not sure about the minimum.

A Former Brilliant Member - 5 years, 1 month ago

Thanks so much, I have been trying to solve this problem for about 3 days. I heard that it may be possible to find the minimum using Lagrange multipliers but I'm not sure.

harry obey - 5 years, 1 month ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$