This is for math wizzes?

Question

Okay, well its the first day of school, our teacher HASNT taught us anything, and she is already giving us homework. There's nothing on the worksheet to help in anyway, it just says to sum up the first 6 numbers, which is 1+2+3+4+5+6, which is 21, but i had to do that in my head. Is there a formula or anyway you can explain how to do that, because on this work sheet, theres another question asking me to sum up the first 100 intergers, which i know is 5050, because i looked it up, but i would actaully like to know how to do this.

Answer 1

Yes, the formula is 1 + 2 + ... + n = n(n+1)/2.  There are couple of ways to derive this.  Your teacher was probably hoping that you'd try to think about this and come up with a strategy of your own.

Imagine if you had made a set of "stairs" by putting one block on the ground, then putting a stack of 2 blocks next to that, and then a stack of 3 blocks next to that, and so on.  If you stop at n columns or "steps", then you'd have 1 + 2 + 3 +  ... + n blocks total.

Now imagine if you took a copy of these "stairs", flipped it over, and put it above the other side, just like in this picture:

http://osteele.com/images/2004/grounded-...

This is a rectangle that's n units long and n+1 units high.  So there are n(n+1) blocks total.  But we only want half of this, so the answer to 1+2+3+...+n is n(n+1)/2.

Answer 2

Let's use 1-10 as an example.

Write out 1+2+3+4+5+6+7+8+9+10

Now fold it between 5 and 6 so the second half

is beneath the first half:

(they would be upside down, but never mind)

1 + 2 + 3 + 4 + 5

10 + 9 + 8 + 7 +  6

Notice how they pair up and each pair adds to 11 ? (which is 10+1)

And there are 5 pairs which is (10 / 2) ?

So you multiply 5 x 11 = 55 and voilà, you have the answer.

Now if it's 1 to 100, the sum is (100+1)

and there are 100/2 pairs.

50 x 101 = 5050.

This is what the young Gauss (perhaps the greatest mathematician

of all time) figured out given the same assignment you were given.

As the story goes, the teacher just wanted to give the class some

busy work, and while the others were occupied adding it up

number by number, young Carl whipped out the answer in a minute

or two, much to the teacher's chagrin.

From here we go the general formula,

Sum (1 to n) = n (n+1) / 2

It also works when n is odd.  In that case you

(n-1)/2  pairs plus a middle number of (n+1)/2

((n-1)/2) * (n+1) + (n+1)/2 and that equal n (n+1)/2.

Answer 3

general formula

n(x1+xn)/2

where

n = number of integers

x1 = first number

xn = last number

100(1+100)/2=5050

2+4+6+8+10=30

5(2+10)/2=30

Answer 4

lets stand in n for number

n=n(n+1)/2

because n+1 lets you find only the next number

the reason you have to divide by two is because you multiply n by n.

so if you take the equation 100(100+1)/2

(1000+100)/2=50 50

and 6(6+1)/2

(42+6)/2=21

Answer 5

The first 100 integers are the numbers 1 through 100.  Here's a neat trick: if you "fold" the series in half and add up the terms in pairs, each pair will add up to 101, and there are 50 such pairs.  What I mean is, if you add the first and last numbers, you get 101 (1 + 100 = 101), then if you add the second and second to last, you also get 101 (2 + 99 = 101).  This will work for all the numbers, ending with (50 + 51 = 101), so you will have 50 "sets" of 101.  Now multiply 101 by 50, and you have 5050.

The general formula for this is n(n + 1)/2, where n is the last number in the set.  So in this case: 100(101)/2 = 10100/2 = 5050

Carl Gauss famously did this as a child when he was set the same problem by a teacher who wanted to keep the class quiet and busy for a while.  The teacher was not pleased.

Answer 6

this is the formula for the sum of numbers in an arithmetic sequence.

n/2(a1+an)

n is the number of terms in the sequence

a1 is the first number in the sequence

an is the last number in the sequence

so for the sum of the first 6 numbers the formula is

6/2(1+6)

6 is the number of terms and the last term, and 1 is the first term

when multiplied out the answer is 21

3(7)=21

this formula works for all arithmetic sequences

Answer 7

hey! its simple....theres a very easy way to do that...when you want to add up a sequence of numbers (i.e. 1,2,3.4.5,....) you can put it in a graphing calculator (ti-83, ti-84) and whatever number you're trying to add up to..so lets say you're trying to add up the first 50 numbers...put it in your calculator as 50!  (literally theres an exclamation point that does this work for you)

:) hope this helps!