Series: $a+b+a+b...+a+b, b>=a$

Consider the series $a+b+a+b+...+a+b+a+b, b >=a$

A general expression for the nth sum of this series is given by:

$n (\frac{a+b}{2}) - (\frac{b-a}{2}(n(mod 2))$ )

When $a=b$ this generalizes to $a \times n$

Can anyone point me to more information about this series? It may be interesting to find an expression for the $nth$ term of $a+b+c+....+a+b+c, a<=b<=c$ and so on for more terms $(a,b,c,...,x,y,z)$ etc.

I came up with this expression myself and realize I might be over complicating something simple. Any information is appreciated, especially with respect to notation.

#NumberTheory #ModularArithmetic #Series

Easy Math Editor

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Math	Appears as
Remember to wrap math in `$` ... `$` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Comments

a,b,c... etc are positive integers

steven kidd - 7 years, 5 months ago

With notation I personally dont really like your use of mods although I understand what you mean. I would write: $U_{n} = n(a+b)/2 + (a-b)/4 + ((a-b)/4)(-1)^{n}$ . Now for the generalisation: There are many better mathematicians on this site than me but I will give you my solution, leaving the proof to you (unless you ask me to explain): So, let k be the number of distinct terms (ie. a+b+c+a+b+c.... would be k= 3), n be the number of terms and we also have i where $i \equiv n \pmod{k}$ , and also $i \ne {0}$ . In this case we will call the distinct numbers $a_{1}$ , $a_{2}$ , $a_{3}$ etc. instead of a, b, c My formula would be:( $a_{1} \times (2n-i)$ + $a_{2} \times (2n-i)$ + $a_{3} \times (2n-i)$ + ... + $a_{i-1} \times (2n-i)$ + $a_{i} \times (2n-i)$ + $a_{i+1} \times (n-i)$ + $a_{i+2} \times (n-i)$ + $a_{i+3} \times (n-i)$ + ... + $a_{k} \times (n-i)$ ) / k. If $i = 0$ then it is trivial.

Josh Rowley - 7 years, 5 months ago

I've been looking back over this and realised I made a few errors. First of all, in the case where $k = 2$ it should be $-1^{n+1}$ instead of $-1^{n}$ . As for the generalisation, I made a fairly stupid misrake, it should be: ( $a_{1} \times (n+k-i)$ + $a_{2} \times (n+k-i)$ + ... + $a_{i} \times (n+k-i)$ + $a_{i+1} \times (n-i)$ + $a_{i+2} \times (n-i)$ + ... + $a_{k} \times (n-i)$ )/ k

Josh Rowley - 7 years, 5 months ago

I like how you got rid of the mod, and it makes me wonder weather the same sort of thing could be done to get rid of a $- (n(mod k))$ term as i had used it in a general sense, (n,k positive integers, n <= k). I'll post back once I get some real time to go over this if I have any further questions, thanks :)

steven kidd - 7 years, 5 months ago

@Steven Kidd – I have no idea how to do it efficiently, but I came up with this off the top of my head:

$n\text{ (mod }k)=\dfrac{k\cdot \text{arg}\left(e^{2\pi i n/k}\right)}{2\pi}$

Feel free to simplify it if you can find a way.

Daniel Liu - 6 years, 11 months ago

Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$

Series: \(a+b+a+b...+a+b, b>=a\)

Comments