How much They have earned

Algebra Level pending

A company has rented the store for 25 days.

The first day they have earned €1 by their store.
The second day they earned €4 by their store.
The third day they earned €9.
And so on until the 25th day.

So the earnings on day $x$ is € $x^2$ .

If they have saved all the money from the first day until the 25th day, how much money they have now?

€5025 €625 €5525 €5050 €325 €525

1 solution

David Agustinus
Jun 2, 2017

1st = €1
2nd = €4
3rd = €9
_ _ _ _ __
25th = €625

so

$1^2 + 2^2 + 3^2 + 4^2 + 5^2 + .... + 25^2$

let's make like this

$1^2 + 2^2 + 3^2 + 4^2 + 5^2 + .... + 25^2$ = $1k + 2k + 3k + 4k + ... 25k$
$1^2 + 2^2 + 3^2 + 4^2 + 5^2 + .... + 25^2$ = $k ( 1 + 2 + 3 + 4 + 5 + 6 + ... + 25)$

Now we wanna find what is k.
Let's do some experiment

If there is 1 number
$1^2 = 1k$
$k = 1$
$k = \frac{3}{3}$

if there are 2 number
$1^2 + 2^2 = 1k + 2k$
$5 = 3k$
$k = \frac{5}{3}$

if there are 3 number
$1^2 + 2^2 + 3^2 = 1k + 2k + 3k$
$14 = 6k$
$k = \frac{7}{3}$

Based of my experiment i can make formula that ...
if there are n number
$k = \frac{2n+1}{3}$

let's go to the problem
$1^2 + 2^2 + 3^2 + 4^2 + 5^2 + .... + 25^2$ = $k ( 1 + 2 + 3 + 4 + 5 + 6 + ... + 25)$

There are 25 number
$k = \frac{2(25)+1}{3}$
$k = \frac{50+1}{3}$
$k = \frac{51}{3}$

We found k

$1^2 + 2^2 + 3^2 + 4^2 + 5^2 + .... + 25^2$ = $k ( 1 + 2 + 3 + 4 + 5 + 6 + ... + 25)$
- $1^2 + 2^2 + 3^2 + 4^2 + 5^2 + .... + 25^2$ = $\frac{51}{3}$ ( 1 + 2 + 3 + 4 + 5 + 6 + ... + 25))
$= \frac{51}{3}$ (325)
$= 5525$
So he has €5525