The arithmetic sequence $1,3,5,7,9, \ldots$ represents the set of odd natural numbers. What is the explicit rule for the sequence? $I_k=1 \checkmark+(k-1) 2$ What is the 200th odd natural number? $a_{200}=\square$
Answer (2)
The explicit rule for the sequence is I k = 1 + ( k − 1 ) 2 .
Substitute k = 200 into the formula: I 200 = 1 + ( 200 − 1 ) 2 .
Calculate I 200 = 1 + ( 199 ) 2 = 1 + 398 = 399 .
The 200th odd natural number is 399 .
Explanation
Understanding the Problem We are given the arithmetic sequence of odd natural numbers: 1 , 3 , 5 , 7 , 9 , … . The explicit rule for this sequence is given as I k = 1 + ( k − 1 ) 2 . We want to find the 200th odd natural number, which corresponds to the 200th term in the sequence.
Substituting k=200 To find the 200th term, we substitute k = 200 into the explicit rule: I 200 = 1 + ( 200 − 1 ) 2
Calculating the 200th Term Now, we simplify the expression: I 200 = 1 + ( 199 ) 2 = 1 + 398 = 399
Final Answer Therefore, the 200th odd natural number is 399.
Examples
Understanding arithmetic sequences is useful in many real-life situations. For example, if you save a fixed amount of money each month, the total amount you've saved over time forms an arithmetic sequence. Similarly, if a theater has rows with an increasing number of seats, the number of seats in each row can form an arithmetic sequence. Knowing how to find a specific term in a sequence helps you predict future savings or determine the number of seats in a particular row.
The explicit rule for the odd number sequence is I k = 1 + ( k − 1 ) ⋅ 2 . To find the 200th odd natural number, we substitute k = 200 into the formula, yielding 399 as the result.
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