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What is an arithmetic sequence?
A sequence where the difference between consecutive terms is constant.
What is a geometric sequence?
A sequence where the ratio between consecutive terms is constant.
Define common difference.
The constant difference between consecutive terms in an arithmetic sequence.
Define common ratio.
The constant ratio between consecutive terms in a geometric sequence.
What are discrete points in the context of sequences?
Isolated points on a graph, representing terms of a sequence, not connected by a continuous line.
What is a sequence?
An ordered list of numbers, where each number has a specific position.
Arithmetic sequence: Formula for the nth term (given initial term).
$a_n = a_0 + dn$
Arithmetic sequence: Formula for the nth term (given a known term).
$a_n = a_k + d(n-k)$
Geometric sequence: Formula for the nth term (given initial term).
$g_n = g_0 * r^n$
Geometric sequence: Formula for the nth term (given a known term).
$g_n = g_k * r^(n-k)$
How to find the 15th term of an arithmetic sequence given a_0 and d?
Use the formula $a_n = a_0 + dn$. Substitute n = 15, a_0 with the initial term, and d with the common difference.
How to find the 9th term of a geometric sequence given g_0 and r?
Use the formula $g_n = g_0 * r^n$. Substitute n = 9, g_0 with the initial term, and r with the common ratio.
How to determine if a sequence is arithmetic?
Check if the difference between consecutive terms is constant. If it is, the sequence is arithmetic.
How to determine if a sequence is geometric?
Check if the ratio between consecutive terms is constant. If it is, the sequence is geometric.
Given two terms in an arithmetic sequence, how do you find the common difference?
Use the formula $a_n = a_k + d(n-k)$ and solve for d.
Given two terms in a geometric sequence, how do you find the common ratio?
Use the formula $g_n = g_k * r^(n-k)$ and solve for r.