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27/03/2018
There’s an algebra quiz tomorrow, and I’m wholly unprepared; however, I’m going to waste more time by organising every formula for this unit (Unit 10, series and their sums) in one post. Actually, this might be helpful for finals.
t(n) = t(1) + r(n - 1)
This formula is used for arithmetic sequences. The “t” stands for the corresponding term of the sequence. The “n” is the number of the term in question (for example, the fourth term, fifth term, etc.). The “r” stands for the ratio of the first term to the second, and so on (for example, if each term is adding 2, then the “r” would be 2). So, for the sequence 0, 2, 4, 6,..., the equation would be: t(n) = 0 + 2(n - 1).
t(n) = t(1)(r - 1)^n
This formula is used for geometric sequences. The “t” again stands for the corresponding term of the sequence, and the “n” again stands for the the number of the term in question. The “r” still stands for the ratio. So, for the sequence 1, 4, 16, 64,..., the equation would be: t(n) = (1)(4)^n.
S(A) = (n(a(1) + a(n))) / 2
This formula is used for the sum of arithmetic sequences, and it’s simple to just refer to it as the “Nathan equation,” as it appears to spell out “natan.” The “n” stands for the number of the term in question. The “a” stands for the term which the corresponding “n” dictates (for example, n = 3 and the third term is 4, so a(n) = 4 while a(1) might equal 0). The “S” simply stands for sum, and the “A” for arithmetic. For the sequence 0, 2, 4, 6,..., the equation would be: S(A) = (n(0 + a(n)))/2.
S(G) = (t(1)(r^n - 1)) / (r - 1)
This formula is used for the sum of geometric sequences. The “t” stands for term. The “r” stands for the ratio, and the “n” stands for the number of the term(s) in question. “G” and “S” simply refer to “geometric” and “sum,” respectively. So, if the sequence is 1, 4, 16, 64,..., the equation would be: S(G) = (1(4^n - 1)) / (4 - 1).Â
Please correct me if I’m wrong on any points, but these are the formulas I’ve been taught in class. If I’m mistaken, it’d be helpful to explain, since I have the assessment soon after the quiz. Hopefully this is accurate enough to help someone!
Types of Sequences
Arithmetic -- a sequence that involves adding or subtracting the same value each time -- example: 1, 3, 5, 7, 9, ...
Geometric -- a sequence that involves multiplying or dividing the same value each time -- example: 1, 2, 4, 8, 16, 32, ...
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Ghee Beom Kim Double Spiral
Stuart Errol Anderson commented:
1, 1, 9, 49, 289, 1681, 9801, 57121, 332929, 1940449, 11309769, 65918161, 384199201, 2239277041, 13051463049, 76069501249, 443365544449, 2584123765441, 15061377048201, 87784138523761, 511643454094369, 2982076586042449, 17380816062160329, 101302819786919521, 590436102659356801
Square of the Pell-Lucas numbers, "numerators of continued fraction convergents to sqrt(2)."
The ratios converge (of course?)
OEIS has 'The ratio a(n+1)/a(n) converges to 3 + 2*sqrt(2). - Richard R. Forberg, Aug 14 2013' and 'a(n) = (((1+sqrt(2))^(2n) + (1-sqrt(2))^(2n)) + 2*(-1)^n)/4 - Lambert Klasen'
And someone found Rick Mawbry's page for this related double spiral...
Which must mean that A090390 must also give a sum for 1/2...
1/(2+1/r)(1+1/r+1/r^2+...)=1/2 where r is the limit ratio.
Oh! That doesn't work because the perfect ratio doesn't make a rectangle!
Of course I had to make in GeoGebra.
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Sigma (aka big E) is for sequences telling you to add up when finished. The number under sigma (n=#) is your starting number when filling out your chart. The top number signifies when you shall stop. The equation on the right is the equation you will plug into the equation. If there is a number on the left hand side after adding up all of your 'answers' you will multiply your addition number by the left hand number.
A Grade 7 Math Guide to Sequences and Series
In this lesson, we will learn about sequences.Â
A sequence is a list of numbers that usually possess a certain pattern.Â
The numbers in the sequence are called terms.Â
We will learn two important types of sequences.Â
One type of sequence is called an arithmetic sequence.Â
An arithmetic sequence is a sequence that has a common difference between two terms.Â
For example, the sequence 1, 3, 5, 7, 9 has a common difference of 2 between two consecutive terms.Â
Another type of sequence is called a geometric sequence.Â
A geometric sequence is a sequence that has a common ratio between two terms.Â
For example, the sequence 2, 1/2, 1/8, 1/32 has a common ratio of 1/4 between two consecutive terms.Â
Watch the video to learn more about sequences. Â
Okay I'm really stumped on this problem can someone help me please
"Christine contributes $100 each month to her 401k. What will be the value of Christine's 401k after the 360th deposit (30 years) if the per annum rate of return is assumed to be 12% compounded monthly? Give your answer correct to the nearest dollar." HELP