Evaluate sum_(k=0)^(9)1000x^(k)=5×60 ×1000

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Identify Geometric Series: The given series is a geometric series with the first term a = 1000 (when k=0) and the common ratio r = x (since each term is x times the previous term). The sum of a finite geometric series can be found using the formula S_n = a(1 - r^n) / (1 - r), where n is the number of terms.Calculate Number of Terms: In this case, the number of terms n is 10 (from k=0 to k=9). We can plug the values into the formula to find the sum: S_10 = 1000(1 - x^10) / (1 - x).Use Sum Formula: We are given that the sum of the series is equal to 5×60×1000. This means that S_10 = 5×60×1000.Set Up Equation: Now we can set up the equation 1000(1 - x^10) / (1 - x) = 5×60×1000 and solve for x.Simplify Right Side: First, we simplify the right side of the equation: 5×60×1000 = 300,000.Divide by 1000: Now the equation is 1000(1 - x^10) / (1 - x) = 300,000. To simplify further, we can divide both sides by 1000, which gives us (1 - x^10) / (1 - x) = 300.Consider x=1: This equation is not straightforward to solve algebraically for x, as it involves a polynomial of degree 10. However, we can notice that if x=1, the denominator becomes zero, which is not allowed. Therefore, x cannot be 1. If x were to be 1, the left side of the equation would be undefined, which cannot equal 300. This suggests that there might be an error in the problem statement or a misunderstanding of the problem as it stands.

Evaluate $\sum_{k=0}^{9}1000x^{k}=5\times60\times1000$

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Evaluate ∑k=091000xk=5×60×1000\sum_{k=0}^{9}1000x^{k}=5\times60\times1000∑k=09​1000xk=5×60×1000

Evaluate $\sum_{k=0}^{9}1000x^{k}=5\times60\times1000$