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Suppose I have following series -

$$1, 2, 4, 7, 11, 16, \dots$$

How can I mathematically represent this series? I can't represent it as AP as d is not constant. I couldn't represent it as GP either.

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1  
You might want to be careful about the difference between series and sequences. A sequence is an ordered list of objects and a series is the sum of terms of a sequence. That is, you might want to make clear whether you are talking about a closed form for the sum of the first $n$ terms (the sum is obviously not convergent) or a closed form for term $n$. –  user50407 Dec 18 '12 at 10:32

2 Answers 2

up vote 1 down vote accepted

$n$-th term $=1 + \frac{n(n-1)}{2}$

Method of difference.

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How did you arrive at this formula? What kind of series is this? Could you give me the name or point me to some resource to learn more about it? –  Kshitiz Sharma Dec 18 '12 at 6:43
    
AFAIK This does have any name as in it is a random series, there are lots of series like this. We use a method called "Method of difference" to solve them. Just Google it out or you can refer to any good reference book. –  KingJames Dec 18 '12 at 6:47
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Check out purplemath.com/modules/nextnumb.htm , where they work out 2, 5, 10, 17, 26,.... –  Mario Carneiro Dec 18 '12 at 6:49

$1+0=1$

$1+1=2$

$2+2=4$

$4+3=7$

$7+4=11$

$11+5=16$

$16+6=22$

$22+7=29$

$29+8=37$

$37+9=46$

$46+10=56$

...............

...............

...............

............... Do you see the pattern now?

Can you see how you can write this as a recurrence?

See Method of Differences.

Regards -A

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+1 Nice demonstration... and link! –  amWhy May 12 '13 at 0:38
    
I am not sure the OP got it, was trying to guide them so they could solve it themselves. Thanks! –  Amzoti May 12 '13 at 0:57

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