The provided reference details how to find the nth term of a linear sequence, which is characterized by adding (or subtracting) a constant value between terms. This is not 'multiplying'. Therefore, the question as stated does not have a single answer based on the provided reference. The question could be asking about two possible things:
- How to find the nth term of a sequence where the difference between each term is found by multiplication, as in a geometric sequence.
- How to find the nth term of an arithmetic sequence that uses a multiplier to reach the next term.
Let's address these possible questions based on the reference:
Geometric Sequences
A geometric sequence has a constant ratio between consecutive terms, not a common difference. So, the reference provided does not directly address how to find the nth term of a geometric sequence. The nth term of a geometric sequence is found using the following formula:
-
an = arn-1
where:
- an is the nth term
- a is the first term
- r is the common ratio (the number you multiply by each time)
- n is the term number
Example:
Consider the sequence 2, 6, 18, 54...
- a (first term) = 2
- r (common ratio) = 6/2 = 3
- So the nth term, an = 2 * 3n-1
Arithmetic Sequences that use a multiplier
The reference discusses how to find the nth term of an arithmetic sequence, where a common difference is added between terms. It mentions multiplying by the common difference, but this doesn't mean the sequence is based on multiplication alone. It means that after finding the common difference, you multiply the term number (n) by the common difference as part of the calculation, which looks like this according to the reference:
- Find the Common Difference (d): Calculate the difference between any two consecutive terms in the sequence.
- Multiply Term Number by Common Difference: Multiply each term number (n = 1, 2, 3...) by the common difference d.
- Find the Constant 'b': Determine the value 'b' by which dn must be adjusted to find the correct terms. This adjustment may involve adding or subtracting some number.
- Write the nth Term: The nth term will be in the form an + b, where 'a' is the common difference 'd' that you calculated, and 'b' is the adjustment you determined above.
The provided reference assumes that sequences are defined by adding the common difference. The 'multiply' instruction is part of calculating the nth term of such a sequence.
Example:
Consider the sequence: 5, 9, 13, 17,...
- The common difference, d, is 4 (9-5 = 4, 13-9=4 etc).
- Multiply the term number (n) by 4: 4n
- Determine the constant 'b'. If n=1, 4n=4, but the actual term is 5, so the constant is 1. Therefore b=1
- The nth term is: 4n + 1
Important Notes
- Linear Sequences: The reference focuses on linear sequences, which have a common difference not a common ratio.
- Geometric vs Arithmetic: It's crucial to first determine if a sequence is arithmetic or geometric.
- No direct "multiply" method: The reference does not describe the method of finding the nth term of a sequence where a fixed number is multiplied to reach the next term.
