Find an explicit formula for the geometric sequence $120, 60, 30, 15, \ldots$. Note: the first term should be $a(1)$. $a(n) =$
Answer (2)
The explicit formula for the geometric sequence 120 , 60 , 30 , 15 , … is given by a ( n ) = 120 ⋅ ( 2 1 ) n − 1 . This formula allows you to find any term in the sequence based on its position. The first term is 120, and the common ratio is \frac{1}{2}.
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Identify the first term: a ( 1 ) = 120 .
Calculate the common ratio: r = 120 60 = 2 1 .
Apply the general formula for a geometric sequence: a ( n ) = a ( 1 ) v er t c d o t r n − 1 .
Substitute the values of a ( 1 ) and r to obtain the explicit formula: a ( n ) = 120 v er t c d o t ( 2 1 ) n − 1 .
The explicit formula for the geometric sequence is a ( n ) = 120 v er t c d o t ( 2 1 ) n − 1 .
Explanation
Identifying the First Term We are given a geometric sequence 120 , 60 , 30 , 15 , … and we want to find an explicit formula for this sequence. The first term is a ( 1 ) = 120 .
Finding the Common Ratio To find the explicit formula for a geometric sequence, we need to determine the common ratio r . The common ratio is the factor by which each term is multiplied to obtain the next term. We can find the common ratio by dividing any term by its preceding term. For example, we can divide the second term by the first term: r = 120 60 = 2 1 = 0.5 We can verify this by dividing the third term by the second term: r = 60 30 = 2 1 = 0.5 And dividing the fourth term by the third term: r = 30 15 = 2 1 = 0.5 So the common ratio is indeed r = 2 1 .
General Formula for Geometric Sequence The general formula for a geometric sequence is given by: a ( n ) = a ( 1 ) ⋅ r n − 1 where a ( n ) is the n -th term, a ( 1 ) is the first term, and r is the common ratio.
Substituting Values into the Formula In our case, a ( 1 ) = 120 and r = 2 1 . Substituting these values into the general formula, we get: a ( n ) = 120 ⋅ ( 2 1 ) n − 1 This is the explicit formula for the given geometric sequence.
Final Answer Therefore, the explicit formula for the geometric sequence is a ( n ) = 120 ⋅ ( 2 1 ) n − 1
Examples
Geometric sequences are useful in many real-world applications, such as calculating compound interest, modeling population growth or decay, and determining the depreciation of assets. For example, if you deposit $1000 in a bank account that earns 5% interest compounded annually, the amount of money in your account each year forms a geometric sequence. Understanding geometric sequences helps you predict future values and make informed financial decisions.
