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September 13th, 2017, 04:09 PM   #1
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Geometric Series Question.

Why is the formula for a geometric series (n+1)a, where the initial term is a, whenever the common ratio is 1?

I would have expected it to be a(n) because the upper limit of the summation is n. So it would be a + a + a +... + a, n times. Not n+1 times.

Why is it n+1?
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September 13th, 2017, 06:06 PM   #2
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It depends on the lower limit. For 0, you get n+1. For 1, you get n. I presume 0 was intended.
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September 13th, 2017, 07:44 PM   #3
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Quote:
Originally Posted by mathman View Post
It depends on the lower limit. For 0, you get n+1. For 1, you get n. I presume 0 was intended.
Wait, I think I see it. The summation is on ar^j, for j = 0,1,2,...,n

If r is 1, then we add a to itself n times, plus a zeroth time. Right? So it's n+1, and not n.
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September 14th, 2017, 04:11 AM   #4
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The "geometric series", by convention, starts with "i= 0", not 1.

$\displaystyle \sum_{i= 0}^n ar^i= a+ ar+ ar^2+ \cdot\cdot\cdot+ ar^n$
counting the "i=0" term there are n+1 terms in the sum so if r= 1 that is a(n+1).

Last edited by skipjack; September 14th, 2017 at 04:18 AM.
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