TransWikia.com

How can I show the quotient of the $k$th partial sums of $sumlimits_{n=1}^{k} n$ and $sumlimits_{n=1}^{k} n^2$ is $frac{3}{2k+1}$?

Mathematics Asked by Danoram on March 8, 2021

I’ve found using pen and paper that any trivial case of the sum of a sequence of integers from $1$ to $k$ divided by the sum of the squares of these integers is equal to $frac{3}{2k+1}$

for example,

$$
begin{split}
frac{1+2+3}{1^{2}+2^{2}+3^{2}} &= frac{3}{7}\
frac{1+2+3 + 4}{1^{2}+2^{2}+3^{2}+4^{2}} &= frac{3}{9}\
frac{1+2+3 + 4 +5}{1^{2}+2^{2}+3^{2}+4^{2}+5^{2}} &= frac{3}{11}\
frac{1+2+3 + 4 +5+6}{1^{2}+2^{2}+3^{2}+4^{2}+5^{2}+6^{2}} &= frac{3}{13}
end{split}
$$

Using a computer I can verify this is true for very large values of $k$ and so I intuitively assume it is true for all values of $k$ but how do I prove this?

I’ve come up with the following equation to represent this as the quotient of two partial sums but don’t know how to get from the left side of the equation to the right.

$$frac{displaystyle sum_{n=1}^k n}{displaystyle sum_{n=1}^k n^2} = frac{3}{2k+1}$$

2 Answers

There are explicit formulas for the partial sum (that can e.g. be shown by induction): $$ sum_{n=1}^k n = frac{k(k+1)}{2} $$ and $$ sum_{n=1}^k n^2 = frac{k(k+1)(2k+1)}{6}. $$ Take the ratio, and you get your formula.

Correct answer by Raoul on March 8, 2021

As we know or we can easily prove, $sum_{n=1}^kn = dfrac{k(k+1)}{2}$

Now expanding, $(n-1)^3 = n^3 - 3n^2 + 3n - 1$

$n^3 - (n-1)^3 = 3n^2 - 3n + 1$

$ sum_{n=1}^k n^3 - (n-1)^3 = 3sum_{n=1}^kn^2 - 3sum_{n=1}^kn + sum_{n=1}^k1$

$ k^3 = 3sum_{n=1}^kn^2 - 3dfrac{k(k+1)}{2} + k$

$ 3sum_{n=1}^kn^2 = k^3 + 3dfrac{k(k+1)}{2} - k$

$ sum_{n=1}^kn^2 = dfrac{k(k+1)(2k+1)}{6}$

Answered by Math Lover on March 8, 2021

Add your own answers!

Ask a Question

Get help from others!

© 2024 TransWikia.com. All rights reserved. Sites we Love: PCI Database, UKBizDB, Menu Kuliner, Sharing RPP