Money Word Problem

Question

I could not seem to be able to solve this money word problem!

Mathew has received 1 dollar the first week, 3 dollars the 2nd week, 5 dollars the 3rd week and so on until he has received a total of 1 000 000 dollars. What amount did he receive the last week?

EDIT

I went onto the detailed solutions page and it had said the following...

Mathew has received 1 + 3 + 5 + ... = 1 000 000. We know that $n^2$ = 1 000 000 and that $n$ = 1 000. He has received a certain amount (1, 3, 5, ...) every week for 1 000 weeks. The amount x received on the 1 000 the week is given by the equation: (1 + x) ÷ 2 = 1 000. This equation becomes 1 + x = 2 000. We find that x = 1 999. The last week, Mathew received $1 999

Could someone please explain to me why $n^2 = 1000000$?

Answer 1

The reason behind this is that $2n+1=Money$ (True as week 1 is 3 dollars, 2 is 5 dollars, etc.) yields every odd number! And it is $n^2$ because to find the total value of the first $n$ odd numbers you must square it. So we can conclude that $n^2=1000000$ and $n=1000$. To reiterate, the value of the first $n$ odd numbers is 1 million, so we get the square and we have the answer! (See https://artofproblemsolving.com/wiki/index.php?title=Proofs_without_words to understand more)

Answer 2

Hint:

You can model the task as $$ S(N) = \sum_{k=1}^N (2 k - 1) = 1 000 000 $$ Your task is to determine the last term $2N-1$.

One way is trial and error (writing a small computer program) or to find a closed solution for the sum, which will feature $N$. $$ S(N) = 1 000 000 $$ Then solve for $N$ and calculate the last term $2N-1$ from it.

Hint:

You can use the linearity of the summation to transform the sum: $$ S(N) = \sum_{k=1}^N (2 k - 1) = 2 \left(\sum_{k=1}^N k \right) - \left( \sum_{k=1}^N 1 \right) = 2 \left( \sum_{k=1}^N k \right) - N $$ So this is now reduced to finding the sum of the first $N$ integers.

Can you take it from here?