Question

1+3+5+7+...+101

Answer 1

so what exactly is the question here? :)

Answer 2

i need to find the sum using gauss approach

Answer 3

Wouldnt you just add them all together or are you doing something else?

Answer 4

do you know what the Gauss approach is?

Answer 5

Consider this equation paired with the same equation written backward: 101 + 99 + 97 + ... + 5 + 3 + 1.

Answer 6

@koryln do you know what the Gauss approach is?

Answer 7

it would be far easier to help you if you responded

Answer 8

most likely^

Answer 9

im sorry i was trying to right out the problem that ttcarol010 said.... i know the approach but cant figure out the answer

Answer 10

@tcarroll010 has laid out the right principal

Answer 11

Gauss noticed that, for an arithmetic series, if you write the series down and then write it again backwards, then the sum of each paired term will be the same

Answer 12

e.g. 1 + 3 + 5 + 7
7 + 5 + 3 + 1
==========
8 + 8 + 8 + 8

Answer 13

use this principal to find the sum of the series you were given

Answer 14

so, in the example I typed, we could say:

1 + 3 + 5 + 7 = half of (4*8) = half of 32 = 16

Answer 15

because 4 * 8 gives you the sum of this series twice - that is why we need to half it

Answer 16

do i need to multiply 101 * 50 then divide it by 2?

Answer 17

take it one step at a time

Answer 18

what would you get if you paired up the last term with the first term?

Answer 19

i.e. what is 101 + 1 = ?

Answer 20

102

Answer 21

good.

now what you need to do next is to work out how many terms there are altogether

Answer 22

do you know how to do that?

Answer 23

51?

Answer 24

perfect!

Answer 25

so - do you know what to do next?

Answer 26

do i multiply that my 102?

Answer 27

remember when we had just 4 terms: 1 + 3 + 5 + 7
the sum of 1st and last term was = 8

and we did: 1 + 3 + 5 + 7 = half of 4 * 8 = 16

Answer 28

now im a little lost...

Answer 29

np - let me try and explain again...

Answer 30

i would multiply by 16?

Answer 31

in your case we have:

1 + 3 + 5 + 7 + ... + 101
101 + 99 + 97 + 95 + ... + 1
=======================
102 + 102 + 102 + 102 + ... + 102

Answer 32

so we end up with how many 102's?

Answer 33

51

Answer 34

correct, so we just multiply 102 by 51
but this will give us twice the sum - so what do we do then?

Answer 35

divide by 2?

Answer 36

for example: 1+3+5 has 3 numbers or it has (1+5)/2 numbers in the expression.
Using the sum of arithmetic sequences we can find the answer.
(1+101)/2 = 56
Therefore, there are 56 numbers in the equation.
Arithemtic sum equation :
Sn = (n(2a+(n-1)d))/2
a is the starting number which is 1
d is the difference or 2
n is the number you want to get up to or 56

So the answer is equal to (56(2+(55)*2))/2 = 3136
Therefore 1 + 3 + 5+...+99 +101 = 3136

Answer 37

2601

Answer 38

thats it - so final answer would be:

1+3+5+7+...+101 = (102 * 51) / 2

Answer 39

perfect!

Answer 40

I hope you understood the principal

Answer 41

I believe Gauss was only 12 when he came up with this! O.O