Sequences and series appear throughout A-Level and AP Precalculus, and this deck of 20 flashcards covers both the arithmetic and geometric cases in full. You'll learn the nth-term and sum formulas for arithmetic sequences, then the equivalent formulas for geom...
Sequences and series appear throughout A-Level and AP Precalculus, and this deck of 20 flashcards covers both the arithmetic and geometric cases in full. You'll learn the nth-term and sum formulas for arithmetic sequences, then the equivalent formulas for geometric sequences, including the special case of an infinite geometric series and the condition for it to converge. The deck also introduces sigma notation, recursive versus explicit formulas, and named sequences including Fibonacci and harmonic sequences, finishing with an introduction to convergence and divergence testing using the ratio test. A dedicated card explains why the harmonic series diverges despite its terms shrinking toward zero — a classic exam trick question. These flashcards pair each formula with a plain-English explanation, helping you understand not just how to calculate a sum but why the underlying pattern works.
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A sequence is an ordered list of numbers (terms). A series is the sum of the terms of a sequence.
𝑎
𝑛
=
𝑎
1
+
(
𝑛
−
1
)
𝑑
a
n
=a
1
+(n−1)d, where
𝑎
1
a
1
is the first term and
𝑑
d is the common difference.
𝑆
𝑛
=
𝑛
2
(
2
𝑎
1
+
(
𝑛
−
1
)
𝑑
)
=
𝑛
2
(
𝑎
1
+
𝑎
𝑛
)
S
n
=
2
n
(2a
1
+(n−1)d)=
2
n
(a
1
+a
n
).
𝑎
𝑛
=
𝑎
1
⋅
𝑟
𝑛
−
1
a
n
=a
1
⋅r
n−1
, where
𝑟
r is the common ratio.
𝑆
𝑛
=
𝑎
1
(
1
−
𝑟
𝑛
)
1
−
𝑟
S
n
=
1−r
a
1
(1−r
n
)
, for
𝑟
≠
1
r
=1.
𝑆
∞
=
𝑎
1
1
−
𝑟
S
∞
=
1−r
a
1
, valid only when $
A series converges if its partial sums approach a finite limit as more terms are added. It diverges if the sum grows without bound or oscillates without settling.
The fixed number by which each term is multiplied to get the next term:
𝑟
=
𝑎
𝑛
+
1
/
𝑎
𝑛
r=a
n+1
/a
n
.
The fixed amount added to each term to get the next term:
𝑑
=
𝑎
𝑛
+
1
−
𝑎
𝑛
d=a
n+1
−a
n
.
A sequence where each term is the sum of the two preceding terms: 0, 1, 1, 2, 3, 5, 8, 13, ...
A concise way to write the sum of a sequence of terms, e.g.
∑
𝑖
=
1
𝑛
𝑖
∑
i=1
n
i means the sum of integers from 1 to n.
