Write variable expressions for arithmetic sequences
key notes:
Arithmetic Sequence:
- An arithmetic sequence is a sequence of numbers in which the difference between consecutive terms is constant. This difference is called the common difference (denoted as d).
- Example: 2,5,8,11,…(common difference d=3.
General Form:
- The n-th term of an arithmetic sequence can be expressed using the formula: an=a1+(n−1)d
Where:
- an = the n-th term
- a1 = the first term of the sequence
- n = the term number
- d = common difference
Example Expression:
- For a sequence starting with 4 and having a common difference of 3:
- a1=4
- d=3
- The expression for the n-th term:
an=4+(n−1)⋅3
Simplifying gives: an=3n+1
Learn with an example
🔔 Write an expression to describe the sequence below.
Use n to represent the position of a term in the sequence, where n = 1 for the first term.
-16, -15, -14, -13, …
The sequence -16, -15, -14, -13, … looks like 1, 2, 3, 4, … except each term is 17 smaller. So, the expression that describes the sequence is n − 17. Check the first four terms:
To find the 1st term, plug in n = 1.
n − 17 = 1 − 17 = –16
To find the 2nd term, plug in n = 2.
n − 17 = 2 − 17 = –15
To find the 3rd term, plug-in n = 3.
n − 17 = 3 − 17 = –14
To find the 4th term, plug-in n = 4.
n − 17 = 4 − 17 = –13
The sequence –16, –15, –14, –13, … is described by the expression n − 17.
🔔 Write an expression to describe the sequence below.
Use n to represent the position of a term in the sequence, where n = 1 for the first term.
-2, -4, -6, -8, …
The sequence -2, -4, -6, -8, … looks like 1, 2, 3, 4, … except each term is multiplied by –2. So, the expression that describes the sequence is –2n, where n represents the position of a term in the sequence. Check the first four terms:
To find the 1st term, plug in n = 1.
–2n = –2(1) = –2
To find the 2nd term, plug in n = 2.
–2n = –2(2) = –4
To find the 3rd term, plug-in n = 3.
–2n = –2(3) = –6
To find the 4th term, plug-in n = 4.
–2n = –2(4) = –8
The sequence –2, –4, –6, –8, … is described by the expression –2n.
🔔 Write an expression to describe the sequence below.
Use n to represent the position of a term in the sequence, where n = 1 for the first term.
58, 59, 60, 61, …
The sequence 58, 59, 60, 61, … looks like 1, 2, 3, 4, … except each term is 57 larger. So, the expression that describes the sequence is n + 57. Check the first four terms:
To find the 1st term, plug in n = 1.
n + 57 = 1 + 57 = 58
To find the 2nd term, plug in n = 2.
n + 57 = 2 + 57 = 59
To find the 3rd term, plug-in n = 3.
n + 57 = 3 + 57 = 60
To find the 4th term, plug-in n = 4.
n + 57 = 4 + 57 = 61
The sequence 58, 59, 60, 61, … is described by the expression n + 57.
let’s practice!🖊️
