Select the correct answer.
Consider the first four terms of the sequence below. $-3,-12,-48,-192, \ldots$
What is the $8^{\text {th }}$ term of this sequence?
A. $-12,288$
B. $-49,152$
C. -768
D. $-196,608
Answer (2)
Determine that the sequence is geometric.
Calculate the common ratio: r = 4 .
Apply the formula for the nth term of a geometric sequence: a n = a 1 ⋅ r n − 1 .
Calculate the 8th term: a 8 = − 3 ⋅ 4 7 = − 49 , 152 .
Explanation
Identifying the Sequence We are given the first four terms of a sequence: − 3 , − 12 , − 48 , − 192 , … . Our goal is to find the 8 th term of this sequence. First, we need to determine the type of sequence.
Finding the Common Ratio To determine if the sequence is arithmetic or geometric, we can check the ratio between consecutive terms. If the ratio is constant, the sequence is geometric. Let's calculate the ratios: − 3 − 12 = 4 − 12 − 48 = 4 − 48 − 192 = 4 Since the ratio between consecutive terms is constant and equal to 4, the sequence is geometric with a common ratio r = 4 .
Applying the Geometric Sequence Formula The general formula for the n th term of a geometric sequence is given by: a n = a 1 ⋅ r n − 1 where a 1 is the first term, r is the common ratio, and n is the term number. In this case, we have a 1 = − 3 , r = 4 , and we want to find the 8 th term, so n = 8 .
Calculating the 8th Term Substituting the values into the formula, we get: a 8 = − 3 ⋅ 4 8 − 1 = − 3 ⋅ 4 7 Now, we calculate 4 7 : 4 7 = 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 ⋅ 4 = 16384 So, a 8 = − 3 ⋅ 16384 = − 49152
Final Answer Therefore, the 8 th term of the sequence is − 49 , 152 .
Examples
Geometric sequences are useful in many real-world applications, such as calculating compound interest, modeling population growth, and determining the depreciation of assets. For example, if you invest $1000 in an account that earns 5% interest compounded annually, the value of your investment each year forms a geometric sequence. Understanding geometric sequences helps you predict how your investment will grow over time.
The 8th term of the sequence − 3 , − 12 , − 48 , − 192 , … can be calculated using the geometric formula. By finding the common ratio and applying the formula, we determined that the 8th term is − 49 , 152 . The correct answer is B. -49,152.
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