Arithmetic Sequence Help

[S_100]

Sum the following series: Image attached

I understand
S_n = n/2(a+l)
and that the following series could be rewritten as
0 + 1 + 2 +...+ n
d= log₂2 = 1
l = n
a=? log₂1 or log₂2

but the part where I am confused is what is the first term,
a= 0 or 1

If it is 1 why does zero not count as a the first term since it is first in the sequence?

with a = 0 I get the sum of the series to be: S_n = n/2 (0+n) = n²/2
but I notice this does work when checked,
So why is it zero is not counted as a first term?



with a = 1 I get S_n = n/2 (1+n)

 

[Dr.Peterson]

You could use your formula to sum either 0 + 1 + 2 +...+ n or 1 + 2 +...+ n, which of course have the same sum.

In the first case (taking it as written), though, there are n+1 terms, not n! So the formula yields (n+1)/2*(0 + n) = n(n+1)/2.

In the second case, the first term is 1 and there are n terms, so the formula is n/2*(1+n), which is the same thing.

When you apply a formula, you have to pay attention to the meaning of the parameters. Here, "n" is not just "whatever has already been called n in the context", but "the number of terms in the arithmetic series".

 

[S_100]

When you apply a formula, you have to pay attention to the meaning of the parameters. Here, "n" is not just "whatever has already been called n in the context", but "the number of terms in the arithmetic series".

Thank you!