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A string is divided into $5$ parts with lengths forming a geometric sequence. If the shortest string is $16$ cm and the longest string is $81$ cm, then the original length of the string is...

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Q. A string is divided into

5

parts with lengths forming a geometric sequence. If the shortest string is

16

cm and the longest string is

81

cm, then the original length of the string is...

Given Information: We are given that the string is divided into $5$ parts with lengths forming a geometric sequence. The shortest part is $16$ cm and the longest part is $81$ cm. In a geometric sequence, each term after the first is found by multiplying the previous term by a constant called the common ratio ( $r$ ). We need to find the common ratio and then use it to find the sum of the sequence, which will give us the original length of the string.
Equation for Longest String: Let's denote the shortest string as $a$ (the first term of the geometric sequence). The longest string will be $a \times r^{(n-1)}$ , where $n$ is the number of terms, which is $5$ in this case. We can set up the equation for the longest string as follows: $\newline$ $a \times r^{(5-1)} = 81 \, \text{cm}$ $\newline$ Since we know the shortest string is $16 \, \text{cm}$ , we can substitute $a$ with $16 \, \text{cm}$ : $\newline$ $16 \times r^4 = 81$
Finding Common Ratio: To find the common ratio $r$ , we need to solve the equation: $\newline$ $16 \cdot r^4 = 81$ $\newline$ Divide both sides by $16$ to isolate $r^4$ : $\newline$ $r^4 = \frac{81}{16}$ $\newline$ $r^4 = 5.0625$ $\newline$ Now, take the fourth root of both sides to find $r$ : $\newline$ $r = (5.0625)^{\frac{1}{4}}$ $\newline$ $r \approx 1.5$
Finding Sum of Geometric Sequence: Now that we have the common ratio, we can find the sum of the geometric sequence using the formula for the sum of the first $n$ terms of a geometric sequence: $S_n = a \times (1 - r^n) / (1 - r)$ where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. We have $a = 16$ cm, $r \approx 1.5$ , and $n = 5$ .
Finding Sum of Geometric Sequence: Now that we have the common ratio, we can find the sum of the geometric sequence using the formula for the sum of the first $n$ terms of a geometric sequence: $S_n = a \cdot \frac{1 - r^n}{1 - r}$ where $S_n$ is the sum of the first $n$ terms, $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. We have $a = 16$ cm, $r \approx 1.5$ , and $n = 5$ . Let's plug the values into the sum formula: $S_5 = 16 \cdot \frac{1 - 1.5^5}{1 - 1.5}$ $S_5 = 16 \cdot \frac{1 - 7.59375}{1 - 1.5}$ $S_5 = 16 \cdot \frac{-6.59375}{-0.5}$ $S_5 = 16 \cdot 13.1875$ \[S_5 = \(211\)\) cm

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