Given $15, 8, 1 \ldots$ in which term is -426?
Answer (2)
Find the common difference: d = 8 − 15 = − 7 .
Use the formula for the n -th term of an arithmetic sequence: a n = a 1 + ( n − 1 ) d .
Substitute a n = − 426 , a 1 = 15 , and d = − 7 into the formula: − 426 = 15 + ( n − 1 ) ( − 7 ) .
Solve for n : n = 64 . Therefore, the answer is 64 .
Explanation
Understanding the Problem We are given an arithmetic sequence 15 , 8 , 1 , … and we want to find which term in the sequence is equal to − 426 .
Finding the Common Difference First, we need to find the common difference d of the arithmetic sequence. We can find this by subtracting consecutive terms.
Calculating the Common Difference The common difference d is calculated as follows: d = 8 − 15 = − 7 So, d = − 7 .
Stating the Formula for the nth Term The first term of the sequence is a 1 = 15 . The formula for the n -th term of an arithmetic sequence is given by: a n = a 1 + ( n − 1 ) d where a n is the n -th term, a 1 is the first term, n is the term number, and d is the common difference.
Setting up the Equation We want to find the term number n such that a n = − 426 . Plugging the values into the formula, we get: − 426 = 15 + ( n − 1 ) ( − 7 ) Now, we solve for n .
Solving for n Subtract 15 from both sides: − 426 − 15 = ( n − 1 ) ( − 7 ) − 441 = ( n − 1 ) ( − 7 ) Divide both sides by − 7 :
− 7 − 441 = n − 1 63 = n − 1 Add 1 to both sides: 63 + 1 = n n = 64 So, the 64th term of the sequence is − 426 .
Final Answer Therefore, the term − 426 is the 64th term in the sequence.
Examples
Arithmetic sequences are useful in many real-life situations, such as calculating simple interest, predicting the depreciation of an asset, or determining the number of seats in rows of a theater. 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 you predict future values and plan accordingly.
The term -426 is the 64th term in the arithmetic sequence starting with 15, 8, 1, ... To find it, we identified the common difference as -7 and used the formula for the n-th term. After solving for n, we found that the answer is 64.
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