Function of N: f(1) = 1, f(2) = 1 + 2 = 3, f(3) = 1 + 2 + 3

[celebrityrose000]

For any natural number "n" , let f(n) denote the sum of the numbers from 1 to n.

Thus, f(1) = 1, f(2) = 1+2 = 3, f (3) = 1+2+3 = 6, f(100) = 1+2+3.....+100 = 5050. It turns out that f is a polynomial of degree 2 in n . Figure out the coefficients of...

f(n)= ___n^2+___n+_____

I'm not sure how to even begin this problem. The only hint the teacher gave us was...

"There is a story about Carl Friedrich Gauss (1777-1855) who may have been the most outstanding mathematician in human history. According to the story, when Gauss was seven years old, his teacher at one stage was unhappy with the class and as a punishment he asked them to compute f(100) . Gauss' class mates started writing the numbers from 1 to 100 on their paper, and adding those numbers. Gauss stared at the ceiling and then wrote the single number 5050 on the sheet and handed it in. You aren't Gauss, but you also aren't seven years old, so maybe you can figure out what he was thinking!
Hint: Think about how to do this in your head for large values of n"


I understand how to do the f(n) = 1 +2 =3. But I don't understand how to find the coefficients of the polynomial thing.

 

[Deleted member 4993]

Re: Functions of N

For any natural number "n" , let f(n) denote the sum of the numbers from 1 to n.

Thus, f(1) = 1, f(2) = 1+2 = 3, f (3) = 1+2+3 = 6, f(100) = 1+2+3.....+100 = 5050. It turns out that f is a polynomial of degree 2 in n . Figure out the coefficients of...

f(n)= ___n^2+___n+_____

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101

How many 101's would you have?

Now look at f(n)

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

How many (n+1) would you get?

What will be the result of their sum?

 

[fasteddie65]

The really depressing part of this problem's history is that Gauss was about 10 years old when he figured it out.