Find the sum of the first 10 terms of the following sequence. Round to the nearest hundredth if necessary. 4,quad-10,quad25,dots Sum of a finite geometric series: S_(n)=(a_(1)-a_(1)r^(n))/(1-r) Answer:

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Find the sum of the first 10 terms of the following sequence. Round to the nearest hundredth if necessary.  4,quad-10,quad25,dots Sum of a finite geometric series:  S_(n)=(a_(1)-a_(1)r^(n))/(1-r) Answer:

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Identify Pattern: First, we need to identify the pattern of the sequence to determine if it's arithmetic, geometric, or neither. By examining the given terms, we can see that each term is multiplied by a common ratio to get the next term: -10 is -2.5 times 4, and 25 is -2.5 times -10. This indicates that the sequence is geometric with a common ratio of -2.5.Find Sum Formula: Next, we need to find the sum of the first 10 terms of this geometric sequence. We can use the formula for the sum of the first n terms of a geometric series, which is:  S_n = (a_1 - a_1 * r^n) / (1 - r)  where S_n is the sum of the first n terms, a_1 is the first term, r is the common ratio, and n is the number of terms.Plug in Values: Now, let's plug in the values we know into the formula. The first term a_1 is 4, the common ratio r is -2.5, and the number of terms n is 10.  S_10 = (4 - 4 * (-2.5)^10) / (1 - (-2.5))Calculate Numerator: We calculate the numerator and the denominator separately. First, we calculate 4 * (-2.5)^10:  4 * (-2.5)^10 = 4 * 9536.7431640625 (rounded to the nearest hundredth)Calculate Denominator: Now we calculate the denominator:  1 - (-2.5) = 1 + 2.5 = 3.5Substitute Values: We substitute these values back into the formula to find S_10:  S_10 = (4 - 38146.97265625) / 3.5Perform Subtraction: Now we perform the subtraction in the numerator:  4 - 38146.97265625 = -38142.97265625Divide Numerator: Finally, we divide the numerator by the denominator to find S_10:  S_10 = -38142.97265625 / 3.5Perform Division: Performing the division gives us:  S_10 ≈ -10900.85 (rounded to the nearest hundredth)

Find the sum of the first $10$ terms of the following sequence. Round to the nearest hundredth if necessary. $\newline$ $4, \quad-10, \quad 25, \ldots$ $\newline$ Sum of a finite geometric series: $\newline$ $S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r}$ $\newline$ Answer:

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