# 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 ∑ar, between r=0 and r = infinity.

## 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'.

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

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

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## Homework Statement

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

## 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,...

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...

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.

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 ar 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.

I'm confused. I want an expression for ar 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.

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?

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.

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!