Find the sum of the first 7 terms of the following sequence. Round to the nearest hundredth if necessary. 21,quad-7,quad(7)/(3),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 7 terms of the following sequence. Round to the nearest hundredth if necessary.  21,quad-7,quad(7)/(3),dots Sum of a finite geometric series:  S_(n)=(a_(1)-a_(1)r^(n))/(1-r) Answer:

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Identify sequence type: To find the sum of the first 7 terms of the given sequence, we first need to identify the type of sequence. The sequence provided is a geometric sequence because each term is obtained by multiplying the previous term by a common ratio (r). To find the common ratio, we divide the second term by the first term.Calculate common ratio: Calculate the common ratio (r) by dividing the second term (-7) by the first term (21). r = -7 / 21 = -1/3Use sum formula: Now that we have the common ratio, we can use the formula for the sum of the first n terms of a geometric series: 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 values into formula: Plug the values into the formula to find the sum of the first 7 terms: S_7 = (21 - 21 * (-1/3)^7) / (1 - (-1/3))Calculate numerator: Calculate the numerator of the formula: 21 * (-1/3)^7 = 21 * (-1/2187) = -21/2187 = -0.00959488...Subtract from first term: Subtract this value from the first term (21): 21 - (-0.00959488...) = 21 + 0.00959488... ≈ 21.01Calculate denominator: Calculate the denominator of the formula: 1 - (-1/3) = 1 + 1/3 = 4/3Divide numerator by denominator: Divide the numerator by the denominator to find the sum of the first 7 terms: S_7 = 21.01 / (4/3) = 21.01 * (3/4) ≈ 15.7575Round to nearest hundredth: Round the result to the nearest hundredth: S_7 ≈ 15.76

Find the sum of the first $7$ terms of the following sequence. Round to the nearest hundredth if necessary. $\newline$ $21, \quad-7, \quad \frac{7}{3}, \ldots$ $\newline$ Sum of a finite geometric series: $\newline$ $S_{n}=\frac{a_{1}-a_{1} r^{n}}{1-r}$ $\newline$ Answer:

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