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th Term Rule
Nth term rule – the equation that allows you to find any term in the sequence
Tn – the term in the “nth” position
n – the position of the term of a sequence
a – the 1st term in a sequence (also T1)
r – the constant ratio between terms in a geometric sequence
d- the constant difference (slope) between terms in an arithmetic sequence
Recursive formula: one that describes the relationship between terms; generates the next
term in the sequence.
Arithmetic sequence – Linear function
A sequence in which the terms differ by a constant. This is a linear relationship.
The formula: y = mx + b
(y = Tn x = n m = constant difference b = T0, the y-intercept)
The nth term rule: Tn = a + d(n-1)
Geometric sequence – Exponential Function
A sequence in which the terms have a constant ratio, an exponential relationship.
The nth term rule: Tn = ar (n-1)
Fibonacci sequence: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55…
Recursive formula: Add the two previous terms to get the next term in the sequence
Triangular numbers: 1, 3, 6, 10, 15, 21, 28, 36, 45, 55…
Recursive formula: The sum of the first n-consecutive numbers.
Nth term rule: Tn = n(n+1)/2
Square numbers: 1, 4, 9, 16, 25, 36, 49, 64…
Recursive formula: Add the next consecutive odd number to generate the next term.
Nth term rule: Tn = n2
Generate the first four terms in each sequence given the nth term rule.
a. Tn = 4n + 1
b. Tn =
n2
2
Find the next three terms in each sequence.
a. 1, 16, 81, 256, ___, ___, ___
b. x, 2x-4, 4x-12, 8x-28, __________, ___________, ____________
c. 4, 12, 14, 42, 44, ___, ____, ____
Write an nth term rule for the following sequences.
a. 11, 13, 15, 17, …
b. 5, 2.5, 1.25, .625, …
Examine the following design. Use t-charts to show your work.
Row 1
Row 2
Row 3
a. If the design were to continue downward, how many squares would there be in the
50th row?
b. What would be the total number of small squares in the first 50 rows?
Consider the following series of equations.
22 − 12 = 3
32 − 22 = 5
42 − 32 = 7
a. If the pattern continues, what is the next equation?
b. Can you find an nth term rule for this series?? Try it! (Helpful hint: remember to
number each “term”—equation, in this case)