#geometric sequence

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). 

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, ...

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!

In this lesson, we will learn about sequences. 

A sequence is a list of numbers that usually possess a certain pattern. 

The numbers in the sequence are called terms. 

We will learn two important types of sequences. 

One type of sequence is called an arithmetic sequence. 

An arithmetic sequence is a sequence that has a common difference between two terms. 

For example, the sequence 1, 3, 5, 7, 9 has a common difference of 2 between two consecutive terms. 

Another type of sequence is called a geometric sequence. 

A geometric sequence is a sequence that has a common ratio between two terms. 

For example, the sequence 2, 1/2, 1/8, 1/32 has a common ratio of 1/4 between two consecutive terms.