Give me a sum to do :-)

[essentialsaltes]

It's one of those anecdotes that's hard to pin down to facts. I'd say it's more likely than Washington cutting down a cherry tree, but I don't think we have any eyewitness testimony from his schoolteacher.

 

[dms1972]

Am I adding it up right?

 

[dms1972]

It's one of those anecdotes that's hard to pin down to facts. I'd say it's more likely than Washington cutting down a cherry tree, but I don't think we have any eyewitness testimony from his schoolteacher.

I thought there might be a formula

 

[dms1972]

there are other aspects to biblical numerology (if that is the correct term) than the number of the beast (though it tends to be the one most speculated about)

 

[essentialsaltes]

Because 5+5 = 10 ???

Mmmm... sort of. (Also, there is a formula. It's just whether this really happened when Gauss was 6 years old is... apocryphal.)

Let's look at it this way. If I had asked you to add seven 8's together:

8+8+8+8+8+8+8, you might not bother adding them up. You might say "the total of seven 8's is just 7 times 8 = 56"

Multiplication is a faster way to do addition.

So what would you have to multiply together to get 55?

 

[Willis Gravning]

Karl Friedrich Gauss' math teacher once asked the class what the sum of 1 + 2 + 3 + ... + 100 was. That's a fine problem to start with.

( n^2 + n)/2 ?

 

[dms1972]

Mmmm... sort of. (Also, there is a formula. It's just whether this really happened when Gauss was 6 years old is... apocryphal.)

oh ok.

Let's look at it this way. If I had asked you to add seven 8's together:

8+8+8+8+8+8+8, you might not bother adding them up. You might say "the total of seven 8's is just 7 times 8 = 56"

Multiplication is a faster way to do addition.

So what would you have to multiply together to get 55?

5 x 11

11 is the next number to be added in the sequence.

 

[dms1972]

1 + 100 = 101
2 + 99 = 101
3 + 98 = 101
4 + 97 = 101

etc.

(reading about it on wikipedia)

 

[Soyeong]

If you add 1 + 99 you get 100, 2 + 98 = 100, etc, until you get to 49 + 51 = 100. So a quick way to add 1-49 and 51-99 would be to do 49 * 100 = 4900. Then just add the 50 in the middle and the 100 at the end for a total of 5050.

 

[dms1972]

can it be put into a formula?

 

[Soyeong]

how can that be put into a formula then?

Willis Gravning already gave the formula:
( n^2 + n)/2

 

[essentialsaltes]

Well let's follow our pattern. You found that 55 was 5*11, which involves "the next one" in the sequence.

When we add the first 20, the 'next one' is 21. And the sum is 210, which is 10*21.

When we add the first 36, the 'next one' is 37. And the sum is 666, which is 18*37.

So you know how to get the 11 and the 21 and the 37. How do you get the 5 or the 10, or the 18?

 

[dms1972]

Willis Gravning already gave the formula:
( n^2 + n)/2

I thought he was asking if that was the formula.

 

[Soyeong]

I thought he was asking if that was the formula.

He was, but he was correct.

 

[Willis Gravning]

I thought he was asking if that was the formula.

I was asking. I came up with it and it seemed to work.

 

[dms1972]

im not sure what

n ^ 2

is, to understand the formula

 

[Willis Gravning]

im not sure what

n ^ 2

is, to understand the formula

n is the whatever the final number in the sequence is. I simplified by reducing the number to 4....1+2+3+4... then worked it out from there. n^2 is n squared.

for 1 to 4 it is 4 squared plus 4 divided by 2 or (16+4)/2=10

for the sequence ending in 100 it is one hundred squared plus one hundred or 10100 divided by 2 = 5050

 

[dms1972]

Well let's follow our pattern. You found that 55 was 5*11, which involves "the next one" in the sequence.

When we add the first 20, the 'next one' is 21. And the sum is 210, which is 10*21.

When we add the first 36, the 'next one' is 37. And the sum is 666, which is 18*37.

So you know how to get the 11 and the 21 and the 37. How do you get the 5 or the 10, or the 18?

divide by 2

10 / 2 = 5
20 / 2 = 10
36 / 2 = 18

which is the last part of the formula

 

[dms1972]

I am not sure my brain is quite capable of 'seeing' or coming up with formulas so quickly. Well done. Its the mathematician gift / skill I suppose to be able to juggle it all in his mind.

 

[essentialsaltes]

Right, so if you're adding up the first 10 numbers, you multiply the next one (11) by half of 10 (5)

If you're adding up the first N numbers, you multiply 'the next one' (N + 1) by 'half of N' (N/2)

(N+1)(N/2)

This is equivalent to Willis' formula.

This can be generalized to all arithmetic sequences (where each number in the sequence goes up by the same amount over the last one)

Then the formula is (first one + last one)(half of how many terms there are)

So to add up the five terms 7 + 9 + 11 + 13 + 15

= (7 + 15)*(5/2)
= (22)*(5/2)
= 55