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

sum_(n=0)^(25)500(0.88)^(n+1)
Answer:

Find the value of the following expression and round to the nearest integer: $\newline$ $\sum_{n=0}^{25} 500(0.88)^{n+1}$ $\newline$ Answer:

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Sum of finite series starts from 1

Full solution

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

\newline

\sum_{n=0}^{25} 500(0.88)^{n+1}

\newline

Answer:

Identify Geometric Series: The given expression is a geometric series, where the first term $a$ is $500 \times 0.88$ , the common ratio $r$ is $0.88$ , and the number of terms $n$ is $26$ (since we start from $n=0$ and go up to $n=25$ ). The sum of a finite geometric series can be calculated using the formula $S = \frac{a(1 - r^n)}{1 - r}$ , where $S$ is the sum of the series.
Calculate First Term: First, calculate the first term of the series, which is $500 \times 0.88$ . $\newline$ $a = 500 \times 0.88 = 440$
Calculate Common Ratio: Next, calculate the common ratio raised to the power of the number of terms, which is $0.88^{26}$ . $\newline$ $r^n = 0.88^{26}$ $\newline$ This calculation can be done using a calculator.
Apply Sum Formula: Now, plug the values of $a$ , $r$ , and $r^n$ into the sum formula for a geometric series. $\newline$ $S = \frac{a(1 - r^n)}{(1 - r)}$ $\newline$ $S = \frac{440(1 - 0.88^{26})}{(1 - 0.88)}$ $\newline$ Again, use a calculator to compute the exact value.
Round Final Answer: After calculating the exact value of the sum, round the result to the nearest integer to get the final answer.

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