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What kind of numbers is in the sequence 1, 8, 27, 64, 125?
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Cubes of integers
Describe the pattern in the sequence 1, 3, 6, 10, 15, 21, 28.
The pattern involves cumulative sums, where each term adds the next integer in order (1, then 2, then 3, and so on).
How is a finite sequence different from an infinite sequence?
A finite sequence has a last term, while an infinite sequence does not, usually indicated by an ellipsis.
What is the fifth term of the sequence given by the general term a_n = (n - 3)^n?
a_5 = (5 - 3)^5 = 2^5 = 32
What steps can be used to identify patterns and formulate general terms in sequences?
1. Analyze differences/similarities between terms 2. Identify repetitious operations 3. Form a general rule based on these observations
What is the definition of a sequence in mathematical terms?
An ordered list of numbers, considered a function whose domain is the set of positive integers.
For the sequence 1, 1/2, 1/3, 1/4, what is the general term?
a_n = 1/n
Derive the general term for the sequence 3, -6, 9, -12, 15.
a_n = (-1)^n * 3n
What is the pattern in the sequence -5, 10, -15, 20, -25?
Multiples of 5 with alternating signs
How do you determine the next terms of an arithmetic sequence?
Add a constant difference to the last term.
What is the next term in the sequence 45, 55, 65?
75
Using the general term a_n = n^3, calculate the 4th term of the sequence.
a_4 = 4^3 = 64
Calculate the general term a_5 of the sequence using the formula a_n = n^2 + n.
a_5 = 5^2 + 5 = 30
What is the general term (a_n) for the sequence of perfect squares?
a_n = n^2
What is the general term for the sequence 1, 3, 6, 10, 15?
a_n = n(n + 1)/2
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