The Final Number

Algebra Level 3

Your Math Prof asks you to write down the Numbers from 1 to 60.

He says, "Add the first two numbers, and place this new number at the end, then erase the first two numbers.

For example, after the first time, the numbers become 3, 4, 5, ..., 59, 60, and 3.

Repeat this process until you get only one number. Whats that last number after you have completed the entire process?"

The other students start the tedious procedure but you are of course smarter than them!

Can you quickly tell your Math Prof what would be The Final Number?

The answer is 1830.

3 solutions

Satyen Nabar
Aug 17, 2014

The Final Number would be the sum of all numbers from 1 to 60.

60*61/2= 1830

Josh Banister
Jan 22, 2015

Let $a$ and $b$ represent the numbers at the beginning of any given step. After a single step of the process has been performed, the numbers $a$ and $b$ are removed and $a+b$ is added. This means that $sum - a - b + (a+b) = sum$ so the sum of the numbers before the step is the same after. By induction, this means the sum of the numbers 1 through 60 is the same as the number that is left which will be $\frac{1}{2}(60)(61) = \boxed{1830}$

Roberto Villadangos Carrera
Jan 22, 2015

It's a classical problem. The fastest way to solve it is to sum first and last, second an second last, and so on... so it gets 61*30=1830

The Final Number

The answer is 1830.

3 solutions

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