What is the sum of the first five terms of the geometric sequence in which a₁ = 3 and r=1/3? Express your answer as an improper fraction using the slash (/) key and no spaces.
Answer (2)
Identify the first term a 1 = 3 and common ratio r = 3 1 .
Apply the formula for the sum of the first n terms of a geometric sequence: S n = a 1 ⋅ 1 − r 1 − r n .
Substitute a 1 = 3 , r = 3 1 , and n = 5 into the formula: S 5 = 3 ⋅ 1 − 3 1 1 − ( 3 1 ) 5 .
Simplify the expression to obtain the sum as an improper fraction: S 5 = 27 121 .
Explanation
Understanding the Problem We are given a geometric sequence with the first term a 1 = 3 and a common ratio r = 3 1 . We want to find the sum of the first five terms of this sequence.
Recall the Sum Formula The formula for the sum of the first n terms of a geometric sequence is given by: S n = a 1 ⋅ 1 − r 1 − r n where a 1 is the first term, r is the common ratio, and n is the number of terms.
Substitute the Values In our case, we have a 1 = 3 , r = 3 1 , and n = 5 . Substituting these values into the formula, we get: S 5 = 3 ⋅ 1 − 3 1 1 − ( 3 1 ) 5
Simplify the Expression Now, let's simplify the expression: S 5 = 3 ⋅ 1 − 3 1 1 − 243 1 = 3 ⋅ 3 3 − 1 243 243 − 1 = 3 ⋅ 3 2 243 242 = 3 ⋅ 243 242 ⋅ 2 3 = 243 ⋅ 2 3 ⋅ 242 ⋅ 3 = 486 2178 We can simplify the fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 18: 486 ÷ 18 2178 ÷ 18 = 27 121 So, the sum of the first five terms is 27 121 .
Final Answer The sum of the first five terms of the geometric sequence is 27 121 .
Examples
Geometric sequences are useful in many real-world scenarios, such as calculating the depreciation of an asset, determining the growth of a population, or modeling compound interest. 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 you predict future values in these types of situations.
To find the sum of the first five terms of the geometric sequence with first term 3 and common ratio 3 1 , we used the sum formula S n = a 1 ⋅ 1 − r 1 − r n . After substituting the values and simplifying, the final answer is 27 121 .
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