Terms of a geometric sequence are found by the formula [tex]T_n=a r^{n-1}[/tex]. If [tex]a=3[/tex] and [tex]r=2[/tex], find the first 4 terms of the sequence.
Answer (2)
The first term is found using the formula T 1 = a = 3 .
The second term is found using the formula T 2 = a r = 3 × 2 = 6 .
The third term is found using the formula T 3 = a r 2 = 3 × 2 2 = 12 .
The fourth term is found using the formula T 4 = a r 3 = 3 × 2 3 = 24 . The first four terms are 3 , 6 , 12 , 24 .
Explanation
Understanding the Problem We are given the formula for the terms of a geometric sequence: T n = a r n − 1 , where T n is the n -th term, a is the first term, and r is the common ratio. We are given a = 3 and r = 2 , and we want to find the first 4 terms of the sequence.
Finding the First Term To find the first term, we substitute n = 1 into the formula: T 1 = a r 1 − 1 = a r 0 = a × 1 = a = 3 So the first term is 3.
Finding the Second Term To find the second term, we substitute n = 2 into the formula: T 2 = a r 2 − 1 = a r 1 = a r = 3 × 2 = 6 So the second term is 6.
Finding the Third Term To find the third term, we substitute n = 3 into the formula: T 3 = a r 3 − 1 = a r 2 = 3 × 2 2 = 3 × 4 = 12 So the third term is 12.
Finding the Fourth Term To find the fourth term, we substitute n = 4 into the formula: T 4 = a r 4 − 1 = a r 3 = 3 × 2 3 = 3 × 8 = 24 So the fourth term is 24.
Listing the Terms Therefore, the first four terms of the geometric sequence are 3, 6, 12, and 24.
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 amounts at the end of each year form a geometric sequence. Understanding geometric sequences helps in making informed financial decisions and predicting future values based on a constant growth rate.
The first four terms of the geometric sequence with initial term a = 3 and common ratio r = 2 are 3, 6, 12, and 24. Each term is calculated using the formula T n = a ⋅ r n − 1 . Following the calculations, we find that the sequence progresses by multiplying by the common ratio each time.
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