I'm teaching myself pre-calculus mathematics. There are plenty of accessible resources on the net showing what the parameters of capital-sigma notation do, but I'm finding there's not enough variation in their examples for me to deduce all the facts...
Let's say there's an arithmetic sequence S of k^2, where the index is k, the lower bound is k = 1, and the upper bound is 2n.
Does that mean we have
S = 2(1^2 + 2^2 + ... + n^2),
or does it mean
S = 1^2 + 2^2 + ... + (2n)^2?
Or does it mean something else?
Also, is my wording correct? I'm particularly unsure of the portion, "there's an arithmetic sequence S of k^2".
Let's say there's an arithmetic sequence S of k^2, where the index is k, the lower bound is k = 1, and the upper bound is 2n.
Does that mean we have
S = 2(1^2 + 2^2 + ... + n^2),
or does it mean
S = 1^2 + 2^2 + ... + (2n)^2?
Or does it mean something else?
Also, is my wording correct? I'm particularly unsure of the portion, "there's an arithmetic sequence S of k^2".
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This is a good question!
That particular sigma notation means the second interpretation you have listed, namely
1^2 + 2^2 + ... + (2n)^2
(a sum with exactly 2n terms in it).
I would read it aloud as "the sum, from k equals one to two n, of k squared" or "the sum of k^2, from k equals one to two n."
If you wanted to indicate 2 (1^2 + 2^2 + ... + n^2), two equivalent ways of doing this would be to write it as
[capital sigma] [k = 1 on the bottom of the sigma] [n on the top of the sigma] 2k^2,
or
2 [capital sigma] [k = 1 on the bottom of the sigma] [n on the top of the sigma] k^2.
This sum would generally not be called the sum of an "arithmetic" sequence, or an "arithmetic series", because successive terms in the sum do not differ by a constant amount. Arithmetic sequences are things like 1, 2, 3, 4, 5 (successive terms differ by the constant 1), or 1, 3, 5, 7, 9 (successive terms differ by the constant 2) or 3, 6, 9, 12, ... (successive terms differ by the constant 3) and so on.
I hope this helped! Sigma notation is a great tool, once you get used to it. It takes a bit of time to learn, but once you get acquainted with it, you can manipulate sums with variable numbers of terms algebraically, without thinking--- sort of like how learning basic algebra lets you manipulate formulas with a fixed number of terms algebraically, without thinking.
That particular sigma notation means the second interpretation you have listed, namely
1^2 + 2^2 + ... + (2n)^2
(a sum with exactly 2n terms in it).
I would read it aloud as "the sum, from k equals one to two n, of k squared" or "the sum of k^2, from k equals one to two n."
If you wanted to indicate 2 (1^2 + 2^2 + ... + n^2), two equivalent ways of doing this would be to write it as
[capital sigma] [k = 1 on the bottom of the sigma] [n on the top of the sigma] 2k^2,
or
2 [capital sigma] [k = 1 on the bottom of the sigma] [n on the top of the sigma] k^2.
This sum would generally not be called the sum of an "arithmetic" sequence, or an "arithmetic series", because successive terms in the sum do not differ by a constant amount. Arithmetic sequences are things like 1, 2, 3, 4, 5 (successive terms differ by the constant 1), or 1, 3, 5, 7, 9 (successive terms differ by the constant 2) or 3, 6, 9, 12, ... (successive terms differ by the constant 3) and so on.
I hope this helped! Sigma notation is a great tool, once you get used to it. It takes a bit of time to learn, but once you get acquainted with it, you can manipulate sums with variable numbers of terms algebraically, without thinking--- sort of like how learning basic algebra lets you manipulate formulas with a fixed number of terms algebraically, without thinking.