Introduction to Sequences
This is the definition of a sequence.
A sequence is an ordered set of numbers.
This is the definition of a series.
An indicated sum of the terms of a sequence.
Or
A sequence of partial sums.
This is the definition of an arithmetic sequence.
A sequence in which successive terms differ by the same number d called the common difference.
Or: A discrete linear function.
This is the definition of a geometric sequence.
A sequence in which the ratio of successive terms is a constant called the common ratio r.
Or: A discrete exponential function.
This is the definition of an infinite geometric series.
A geometric series with infinitely many terms.
These are the values of the first 5 terms of the sequence an = 2n - 3.
a1 = -1, a2 = 1, a3 = 5, a4 = 13, a5 = 29
This is the technical term for the position of a term in the sequence or which sum from the series.
This is the definition of an arithmetic series.
The indicated sum of the terms of an arithmetic sequence.
This is the definition of a geometric series.
The indicated sum of the terms of a geometric sequence.
For these values of the common ratio r, an infinite geometric series converges.
Additionally, for these alternative values of r, and infinite geometric series diverges.
If |r| < 1, then the series is said to converge to a limit. If |r| > 1, then the series is said to diverge.
This is the value of the 5th term of the sequence with a1 = 5 and an = 2an-1 + 1.
a5 = 95
This is the value of S4 of the sequence 1/2, 1/4, 1/8, 1/16, ...
15/16
This is whether the sequence -3, 2, 7, 12, 17, ... is arithmetic.
Additionally, if it is, this is the common difference d.
Arithmetic, d = 5
This is the value of the 10th term of the geometric sequence with a5 = 96 and a7 = 384.
a10 = 3072
This is whether the geometric series 20 + 24 + 28.8 + 34.56 + ... converges or diverges.
Since r = 1.2, the series diverges.
This is the value of the 10th term of the Fibonacci Sequence.
a10 = 55
If $100 is deposited into an account monthly with a 2% annual interest rate, this is how much money will be in the account after 10 years.
$13,271.97, or
About $13,300
Find the 6th term of the arithmetic sequence with
a9 = 120 and a14 = 195.
a6 = 75
This is how long it will take for a principal of $10,000, compounded monthly with a 1.5% annual interest rate, to reach a balance of $11,000.
About 76.3 months
OR: 6.36 years
If it exists, this is the sum of the infinite geometric series 5 + 4 + 3.2 + 2.56 + ...
S = 25
This is a possible explicit definition for the nth term of the sequence 3, 6, 12, 24, 48, ...
an = 3 × 2n-1
This is the only possible infinite arithmetic series that converges.
an = 0
0, 0, 0, 0, ...
This is whether the sequence 6, 10, 15, 21, ... is geometric.
Additionally, if it is, this is the value of r.
The sequence 6, 10, 15, 21, ... is neither geometric nor arithmetic.
This is the value of S7 for the geometric series 3 - 6 + 12 - 24 + ...
S7 = 129
If the second term of a geometric sequence is 5/54 and the fifth term of it is 625/11,664, this is the value of the sequence's infinite series.
S = 2/3