Express a series in sigma notation

[whatisreality]

Homework Statement

I've been given the series 3,4,6,10,18... and asked to express as the ∑a_r, between r=0 and r = infinity.

Homework Equations

The Attempt at a Solution

Well, I can see a pattern! The difference between terms doubles every time. I'm having difficulty expressing this mathematically though... I think it should be 3 plus some function of r. Is there a technique for this? All we've been told is to do it 'by inspection'.

 

[Brian T]

You're on the right track. It goes up by 1, then 2, 4, 8. What pattern does that seem like? Can you think of a function that increments in that manner

 

[whatisreality]

2^r might work?
So that would make the series 3+2^r, except then it doesn't work for r=0...
Maybe 2+2^r instead. That works! :)

 

[Ray Vickson]

Homework Statement

I've been given the series 3,4,6,10,18... and asked to express as the ∑a_r, between r=0 and r = infinity.

Homework Equations

The Attempt at a Solution

Well, I can see a pattern! The difference between terms doubles every time. I'm having difficulty expressing this mathematically though... I think it should be 3 plus some function of r. Is there a technique for this? All we've been told is to do it 'by inspection'.

The differences double, so the terms are 3, 3+1, 3+1+2, 3+1+2+4, 3+1+2+4+8,...

 

[haruspex]

The differences double, so the terms are 3, 3+1, 3+1+2, 3+1+2+4, 3+1+2+4+8,...

That's doing it the hard way! Reminds me of that story about John Nash solving the back-and-forth fly problem by summing the series...

 

[Ray Vickson]

That's doing it the hard way! Reminds me of that story about John Nash solving the back-and-forth fly problem by summing the series...

Might not be the hard way if the question was not really stated correctly by the OP; have we not seen that many times before in this forum? If the OP gave the correct statement, then, of course, you are right; but if the question really wanted an expression for $a_r$ in sigma notation, then what I outlined would go part way to the solution.

 

[whatisreality]

Might not be the hard way if the question was not really stated correctly by the OP; have we not seen that many times before in this forum? If the OP gave the correct statement, then, of course, you are right; but if the question really wanted an expression for $a_r$ in sigma notation, then what I outlined would go part way to the solution.

I'm confused. I want an expression for a_r using sigma notation. I thought that's what my question asked. What does my question actually mean then? As in, I thought I stated the problem correctly, AND the problem was I want an expression using sigma notation.

 

[Ray Vickson]

I'm confused. I want an expression for a_r using sigma notation. I thought that's what my question asked. What does my question actually mean then? As in, I thought I stated the problem correctly, AND the problem was I want an expression using sigma notation.

Well, if you want to express the general ($r$th) term $a_r$ of your sequence in sigma notation, the method I suggested in Post # 4 points the way. Just try to translate that into sigma notation.

 

[Svein]

I've been given the series 3,4,6,10,18... and asked to express as the ∑ar, between r=0 and r = infinity.

Asked to express what as the ∑ar?

 

[haruspex]

Might not be the hard way if the question was not really stated correctly by the OP; have we not seen that many times before in this forum? If the OP gave the correct statement, then, of course, you are right; but if the question really wanted an expression for $a_r$ in sigma notation, then what I outlined would go part way to the solution.

Yes, it's not very clear. Because it referred to the given numbers as a series (which to me implies summation), not a sequence, I was looking for a general closed form for the numbers $a_0 = 3, a_1=4,$... so that the series sum can be written $\Sigma a_r$.
But you are interpreting it as a sequence, which must then 'unsummed' into a series. $a_0 = 3, a_0+a_1=4, a_0+a_1+a_2=6$... That makes it seem a strange question to me, quite apart for the terminology.

 

[whatisreality]

I'm not entirely sure how to differentiate between those two cases. I think I meant the first option though, as in a₀=3, a₁=4 etc. Sorry for a lack of clarity!