To determine if a number belongs to an nth term sequence, you need the formula for the nth term of that sequence. Substitute the number in question into the formula and solve for 'n'. If 'n' is a positive integer, the number is in the sequence; otherwise, it's not.
Understanding nth Term Sequences
An nth term sequence is a mathematical pattern where each term can be calculated using a formula involving its position (n) in the sequence. For example, in an arithmetic sequence (where the difference between consecutive terms is constant), the nth term is often represented as a_n = a_1 + (n - 1)d, where a_1 is the first term and d is the common difference. Other sequences, like geometric or quadratic sequences, have different formulas.
Steps to Check if a Number is in the Sequence
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Identify the nth term formula: You must know the formula that defines the sequence. This might be given explicitly, or you may need to derive it from the sequence's pattern.
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Substitute the number: Let the number you want to check be 'x'. Substitute 'x' into the nth term formula in place of
a_n. -
Solve for 'n': Solve the resulting equation for 'n'. This often involves algebraic manipulation.
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Check if 'n' is a positive integer: If 'n' is a positive whole number, then the number 'x' is part of the sequence at the nth position. If 'n' is not a positive integer (e.g., a fraction, a negative number, or an imaginary number), the number 'x' is not in the sequence.
Examples
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Arithmetic Sequence: Let's say the nth term of an arithmetic sequence is given by
a_n = 3n + 2. We want to know if the number 17 is in this sequence.- Substitute: 17 = 3n + 2
- Solve: 15 = 3n => n = 5
- Conclusion: Since n = 5 (a positive integer), 17 is the 5th term in the sequence.
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Non-Arithmetic Sequence: Consider a sequence where
a_n = n². Is 25 in this sequence?- Substitute: 25 = n²
- Solve: n = ±5
- Conclusion: While we find a solution for n, only the positive integer solution (n=5) is relevant. Therefore, 25 is the 5th term in the sequence.
Important Note: The method depends entirely on knowing or being able to find the correct nth term formula for the specific sequence. Different types of sequences will have different formulas. For some complex sequences, finding the nth term formula itself might be a challenging task.
