In the sequence $6, 11, 16, \ldots$, which term is 236?
Answer (2)
Identify the sequence as arithmetic with first term a 1 = 6 and common difference d = 5 .
Use the formula for the n -th term of an arithmetic sequence: a n = a 1 + ( n − 1 ) d .
Set a n = 236 and solve for n : 236 = 6 + ( n − 1 ) 5 .
Find n = 47 , so the 47th term is 236. 47
Explanation
Understanding the Problem We are given the arithmetic sequence 6 , 11 , 16 , … and we want to find which term is equal to 236.
Finding the Common Difference First, we need to find the common difference d of the sequence. We can find this by subtracting consecutive terms. d = 11 − 6 = 5 . So the common difference is 5.
General Formula The general formula for the n -th term of an arithmetic sequence is given by a n = a 1 + ( n − 1 ) d , where a 1 is the first term and d is the common difference. In this case, a 1 = 6 and d = 5 .
Setting up the Equation We want to find the term number n such that a n = 236 . So we set up the equation: 236 = 6 + ( n − 1 ) 5
Solving for n Now we solve for n :
236 = 6 + 5 ( n − 1 ) 230 = 5 ( n − 1 ) 5 230 = n − 1 46 = n − 1 n = 46 + 1 n = 47
Final Answer Therefore, the 47th term of the sequence is 236.
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 fixed number of additional seats. For example, if you deposit a fixed amount 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.
The 47th term of the arithmetic sequence 6 , 11 , 16 , … is 236. The common difference in the sequence is 5, and the formula used to find the term number is a n = a 1 + ( n − 1 ) d . Thus, solving for n gives us 47.
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