𝑢!"−
The value of the first term
𝑢!"
serves as the starting point for the sequence. It sets the
baseline for the subsequent terms.
•
𝑑 −
The common difference
𝑑,"
determines the rate of change between consecutive terms.
Adding
𝑑
to each term generates the sequence.
•
𝑛 −
The term number
"𝑛"
identifies the position of the term we are interested in within the
sequence. It ranges from 1 for the first term to any positive integer for subsequent terms.
•
𝑢"#$"
The
𝑛
-th term
𝑢"#
is the value of the term located at position
𝑛"
in the sequence. It is
calculated using the formula
𝑢"= 𝑢!+
(
𝑛 − 1
)
𝑑
, which takes into account the first term, the
term number, and the common difference.
•
In this written investigation I will be explaining the findings I discovered when experimenting with
Arithmetic Sequences and finding how one would find
𝑢!, 𝑢",𝑎𝑛𝑑&
u_4&in terms of
𝑢#
and
𝑑
when
looking at an arithmetic sequence with the first term
𝑢#
and a common difference
𝑑
And then
finding an expression for
𝑢$
in terms of
𝑢#, 𝑛, 𝑎𝑛𝑑&𝑑.&
An Arithmetic sequence can be defined as a
type of sequence in mathematics where each term is obtained by adding a constant value (known
as the common difference) to the previous term. This constant difference ensures that the
difference between any two consecutive terms remains the same throughout the sequence. A
sequence is essentially an ordered collection of numbers, they can be finite or infinite and are
usually defined by a formula or recurrence relation that dictates how each term is obtained from
the previous terms. There are four major variables used when defining sequences.
In an arithmetic sequence, these variables work together to define and describe the progression of the
sequence. I created the sequence
10,8,6,4,2,0, −2, −4"
using that as my experienmental sequence. With my
goal in mind I first write
𝑢%, 𝑢&, 𝑢'
in terms of u and d.
•
u2: The second term can be obtained by adding the common difference d to the first term u1
•
𝑢2 =𝑢1 + 𝑑 = 10 + 𝑑
•
U3: The third term is derived by adding d to the second term.
•
𝑢3 =𝑢2 + 𝑑 = 10 + 𝑑 + 𝑑 = 10 +2𝑑
•
u4: The fourth term is obtained by adding d to the third term.
•
𝑢4 =𝑢3 + 𝑑 = 10 +2𝑑 + 𝑑 = 10 +3𝑑
•To find a general expression for the n-th term un, we can observe that for each subsequent term,
we add d an additional n-1 times.
•So, 𝑢𝑛 =𝑢1 + (𝑛 − 1)𝑑
I'd say the consistent pattern made it easier to identify the common difference and establish
expressions for specific terms. However, attentiveness was needed while handling negative terms to
ensure accurate results.
Monday, August 28, 2023
2:09 PM
