Answer:
[tex]\sum\limits^{n=6}_{n=1} {(-4+8n)}[/tex] is the notation to represent the sum of first six terms of the sequence 4, 12, 20, ....
So, the option "the summation from n equals 1 to 6 of negative 4 plus 8 times n" is the correct option as: [tex] \sum\limits^{n=N}_{n=1} {(-4+8n)}[/tex].
Step-by-step explanation:
Let us consider the sequence
4, 12, 20, ....
As d = 12-4 = 20 - 12 = 8
Hence, the sequence is an arithmetic as the common difference between consecutive terms is same (constant).
So,
For the arithmetic sequence,
a, a + d, a + 2d, a + 3d.........a + (n - 1)d
The nth term = a + (n - 1)d
a = first term, d = common difference
For the given sequence 4, 12, 20,...
a = 4, d = 8
Lets put these values in Summation Formula.
Sigma notation to represent the sum of the first six terms of the following sequence: [tex] \sum\limits^{n=N}_{n=1} {a+(n-1)d}[/tex]
⇒ [tex] \sum\limits^{n=N}_{n=1} {4+(n-1)8}[/tex]
⇒ [tex] \sum\limits^{n=N}_{n=1} {(4+8n-8)}[/tex]
⇒ [tex] \sum\limits^{n=N}_{n=1} {(-4+8n)}[/tex]
Hence, [tex]\sum\limits^{n=6}_{n=1} {(-4+8n)}[/tex] is the notation to represent the sum of first six terms of the sequence 4, 12, 20, .....
So, the option "the summation from n equals 1 to 6 of negative 4 plus 8 times n" is the correct option as: [tex] \sum\limits^{n=N}_{n=1} {-4+8n}[/tex].
Keywords: sigma notation, arithmetic sequence, series
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