# Arithmetic Sequences

• Jun 14th 2008, 04:41 PM
damaster
Arithmetic Sequences
This is a question I really don't get. I've tried all the things i can think of, and the only clue i have is: Arithmetic Sequences (this is a topic I'm horrible at).

The problem is

Find the sum of all positive integers not greater than 10000 that are divisible by either 3 or 11 but not by both of them.'
Tkx
Mikhael :)
• Jun 14th 2008, 04:44 PM
sean.1986
It's all the numbers divisible by 3 or 11 but NOT divisible by 33. Does that help?
• Jun 14th 2008, 04:48 PM
sean.1986
So...

S3333, a = 3, d = 3

S909, a = 11, d = 11

S303, a = 33, d = 33
• Jun 14th 2008, 06:46 PM
Soroban
Hello, Mikhael!

We need the formula for the sum of an arithemtic series.

The sum of the first n terms is: .$\displaystyle S_n \:=\:\frac{n}{2}\bigg[2a + (n-1)d\bigg]$

. . where: $\displaystyle a$ = first term, $\displaystyle d$ = common difference.

Quote:

Find the sum of all positive integers not greater than 10000
that are divisible by either 3 or 11 but not by both of them.

Sum of multiples of 3: .$\displaystyle S_3 \;=\; 3 + 6 + 9 + \hdots + 9999$
This is an arithmetic series with: .$\displaystyle a = 3,\;\; d = 3,\;\; n = 3333$
Its sum is: .$\displaystyle S_3 \:=\:\frac{3333}{2}\bigg[2(3) + 3332(2)\bigg] \:=\:16,668,333$

Sum of multiples of 11: .$\displaystyle S_{11} \;=\;11 + 22 + 33 + \hdots + 9999$
This is an arithmetic series with: .$\displaystyle a = 11,\;\;d=11\;\;n = 909$
Its sum is: .$\displaystyle S_{11} \;=\;\frac{909}{2}\bigg[2(11) + 908(11)\bigg] \;=\;4,549,545$

Sum of multiples of 33: .$\displaystyle S_{33} \;=\;33 + 66 + 99 + \hdots + 9999$
This is an arithmetic series with: .$\displaystyle a = 33,\;\;d = 33,\; \;n = 303$
Its sum is: .$\displaystyle S_{33} \;=\;\frac{909}{2}\bigg[2(33) + 302(33\bigg] \;=\;1,519,848$

The desired sum is: .$\displaystyle S_3 + S_{11} - 2(S_{33}) \;=\;\boxed{18,178,182}$

• Jun 14th 2008, 07:46 PM
sean.1986
Shouldn't that be 303/2 for the 3rd one?

And why do you subtract twice the sum of the multiples of 33?