Find the nth term of the sequence 11, 8, 5, 2, -1.
Understand the Problem
The question is asking to find the nth term of the given sequence: 11, 8, 5, 2, -1. To solve this, we will need to analyze the pattern in the sequence to derive a formula for the nth term.
Answer
The formula for the nth term is $a_n = 14 - 3n$.
Answer for screen readers
The nth term of the sequence is given by the formula:
$$ a_n = 14 - 3n $$
Steps to Solve
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Identify the pattern in the sequence
The given sequence is: ( 11, 8, 5, 2, -1 ).
We can observe that each term decreases by 3.
The differences are:
( 11 - 8 = 3 )
( 8 - 5 = 3 )
( 5 - 2 = 3 )
( 2 - (-1) = 3 ) -
Form a general formula
Since the sequence decreases consistently, we can express the nth term as a linear function:
Let the nth term ( a_n ) be represented as:
$$ a_n = a_1 + (n - 1) \cdot d $$
where ( a_1 = 11 ) (the first term) and ( d = -3 ) (the common difference). -
Substitute values into the formula
Substituting the known values into the formula gives:
$$ a_n = 11 + (n - 1)(-3) $$ -
Simplify the formula
Now, simplify the equation:
$$ a_n = 11 - 3(n - 1) $$ $$ = 11 - 3n + 3 $$ $$ = 14 - 3n $$
The nth term of the sequence is given by the formula:
$$ a_n = 14 - 3n $$
More Information
This formula allows us to find any term in the sequence. For example:
- For ( n = 1 ): ( a_1 = 14 - 3 \cdot 1 = 11 )
- For ( n = 2 ): ( a_2 = 14 - 3 \cdot 2 = 8 )
- For ( n = 5 ): ( a_5 = 14 - 3 \cdot 5 = -1 )
This shows how the formula accurately represents the sequence.
Tips
- Not recognizing the constant difference: It's essential to identify that the sequence decreases consistently by the same amount.
- Incorrectly applying the formula for the nth term: Ensure the correct substitution of values into the linear equation format.
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