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Questions and Answers
What is the common difference and the first term in the arithmetic sequence 2, 5, 8, 11?
The common difference is 3, and the first term is 2.
What are the next three terms in the sequence 2, 5, 8, 11?
The next three terms are 14, 17, and 20.
Calculate u0, u1, and u2 for the sequence defined by un = 2 - 3n.
u0 = 2, u1 = -1, and u2 = -4.
How can you prove that the sequence (un)n∈ℕ defined by un = 2 - 3n is arithmetic?
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Is the sequence (wp)p∈ℕ defined by wp = (p+1)² arithmetic, and what are w0, w1, and w2?
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Study Notes
Arithmetic Sequences
- An arithmetic sequence is a sequence where the difference between consecutive terms is constant.
- This constant difference is called the common difference.
- If the common difference (
r) is 0, all terms in the sequence are equal to the first term. - To find the next term in an arithmetic sequence, add the common difference to the current term.
Example Arithmetic Sequence
- The sequence 2, 5, 8, 11,... is arithmetic.
- The common difference (
r) is 3. - The first term is 2.
- Three more terms in the sequence are 14, 17, and 20.
Sequence Defined by Formula
- The sequence (
un) whereun = 2 - 3nis arithmetic. -
u0 = 2,u1 = -1, andu2 = -4. - To prove the sequence is arithmetic, show that the difference between any two consecutive terms is constant:
-
un+1 - un = (2 - 3(n+1)) - (2 - 3n) = -3, which is constant.
-
Non-Arithmetic Sequence
- The sequence (
wp) wherewp = (p+1)^2is not arithmetic. -
w0 = 1,w1 = 4, andw2 = 9. - The difference between consecutive terms is not constant:
-
w1 - w0 = 3 -
w2 - w1 = 5
-
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Description
Test your understanding of arithmetic sequences with this quiz. Explore key concepts such as common differences and sequence definitions through example problems. Determine whether given sequences are arithmetic or not.
