In the arithmetic sequence $10, 13, 16, \ldots$, which term is 247?
In the sequence $6, 11, 16, \ldots$, which term is 236?
Given $15, 8, 1, \ldots$, which term(s) is -426?
Answer (2)
Identify the first term ( a ) and common difference ( d ) for each arithmetic sequence.
Use the formula for the nth term of an arithmetic sequence: a n = a + ( n − 1 ) d .
Substitute the given values into the formula and solve for n .
The term numbers are: 80, 47, and 64. 80 , 47 , 64
Explanation
Understanding the Problem We are given three arithmetic sequences and asked to find which term in each sequence equals a specific value. We will use the formula for the nth term of an arithmetic sequence, which is given by a n = a + ( n − 1 ) d , where a n is the nth term, a is the first term, n is the term number, and d is the common difference.
Analyzing the First Sequence For the first sequence, 10 , 13 , 16 , … , we have a = 10 and d = 13 − 10 = 3 . We want to find the term number n such that a n = 247 . Using the formula, we have: 247 = 10 + ( n − 1 ) 3
Solving for n in the First Sequence Solving for n :
247 − 10 = ( n − 1 ) 3 237 = 3 ( n − 1 ) 3 237 = n − 1 79 = n − 1 n = 79 + 1 = 80 So, 247 is the 80th term in the first sequence.
Analyzing the Second Sequence For the second sequence, 6 , 11 , 16 , … , we have a = 6 and d = 11 − 6 = 5 . We want to find the term number n such that a n = 236 . Using the formula, we have: 236 = 6 + ( n − 1 ) 5
Solving for n in the Second Sequence Solving for n :
236 − 6 = ( n − 1 ) 5 230 = 5 ( n − 1 ) 5 230 = n − 1 46 = n − 1 n = 46 + 1 = 47 So, 236 is the 47th term in the second sequence.
Analyzing the Third Sequence For the third sequence, 15 , 8 , 1 , … , we have a = 15 and d = 8 − 15 = − 7 . We want to find the term number n such that a n = − 426 . Using the formula, we have: − 426 = 15 + ( n − 1 ) ( − 7 )
Solving for n in the Third Sequence Solving for n :
− 426 − 15 = ( n − 1 ) ( − 7 ) − 441 = − 7 ( n − 1 ) − 7 − 441 = n − 1 63 = n − 1 n = 63 + 1 = 64 So, -426 is the 64th term in the third sequence.
Final Answer Therefore, 247 is the 80th term in the first sequence, 236 is the 47th term in the second sequence, and -426 is the 64th term in the third sequence.
Examples
Arithmetic sequences are useful in many real-life situations, such as calculating simple interest, predicting patterns, and determining the number of seats in an auditorium where each row has a constant increase in the number of seats. For example, if you deposit a fixed amount of money into a savings account each month, the total amount in your account over time forms an arithmetic sequence. Understanding arithmetic sequences helps in financial planning and forecasting.
In the first sequence, the 80th term is 247. In the second sequence, the 47th term is 236. In the third sequence, the 64th term is -426.
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