#sum of 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).