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Find the value of the following expression and round to the nearest integer:

sum_(n=1)^(21)100(0.96)^(n-1)
Answer:

Find the value of the following expression and round to the nearest integer: $\newline$ $\sum_{n=1}^{21} 100(0.96)^{n-1}$ $\newline$ Answer:

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Evaluate definite integrals using the chain rule

Full solution

Q. Find the value of the following expression and round to the nearest integer:

\newline

\sum_{n=1}^{21} 100(0.96)^{n-1}

\newline

Answer:

Given values and formula: We are asked to evaluate a geometric series with the first term $a = 100$ and the common ratio $r = 0.96$ . The sum of the first $n$ terms of a geometric series is given by the formula $S_n = \frac{a(1 - r^n)}{(1 - r)}$ , where $n$ is the number of terms. In this case, we want to find the sum of the first $21$ terms.
Calculate power of $0.96^{21}$ : First, let's plug in the values into the formula for the sum of a geometric series: $S_{21} = 100(1 - 0.96^{21}) / (1 - 0.96)$ .
Substitute values into formula: Now, we calculate the power of $0.96^{21}$ using a calculator.
Perform subtraction in numerator: After calculating, we find that $0.96^{21}$ is approximately $0.45289$ .
Perform subtraction in denominator: We substitute this value back into the formula: $S_{21} = \frac{100(1 - 0.45289)}{(1 - 0.96)}$ .
Divide numerator by denominator: Now, we perform the subtraction in the numerator: $1 - 0.45289 = 0.54711$ .
Perform division and multiplication: Next, we perform the subtraction in the denominator: $1 - 0.96 = 0.04$ .
Round to nearest integer: We now divide the numerator by the denominator: $S_{21} = 100 \times 0.54711 / 0.04$ .
Round to nearest integer: We now divide the numerator by the denominator: $S_{21} = 100 \times 0.54711 / 0.04$ . Performing the division and multiplication, we get $S_{21} = 100 \times 13.6775$ .
Round to nearest integer: We now divide the numerator by the denominator: $S_{21} = 100 \times 0.54711 / 0.04$ . Performing the division and multiplication, we get $S_{21} = 100 \times 13.6775$ . Multiplying $100$ by $13.6775$ gives us $S_{21} = 1367.75$ .
Round to nearest integer: We now divide the numerator by the denominator: $S_{21} = 100 \times 0.54711 / 0.04$ . Performing the division and multiplication, we get $S_{21} = 100 \times 13.6775$ . Multiplying $100$ by $13.6775$ gives us $S_{21} = 1367.75$ . Finally, we round $1367.75$ to the nearest integer, which is $1368$ .

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