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Find the sum of the first 6 terms of the following series, to the nearest integer.

20,10,5,dots
Answer:

Find the sum of the first $6$ terms of the following series, to the nearest integer. $\newline$ $20,10,5, \ldots$ $\newline$ Answer:

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Geometric sequences

Full solution

Q. Find the sum of the first

6

terms of the following series, to the nearest integer.

\newline

20,10,5, \ldots

\newline

Answer:

Identify type and ratio: Identify the type of series and the common ratio. $\newline$ The given series is geometric because each term is obtained by multiplying the previous term by a constant ratio. $\newline$ To find the common ratio $r$ , we divide the second term by the first term. $\newline$ $r = \frac{10}{20} = 0.5$
Use formula for sum: Use the formula for the sum of the first $n$ terms of a geometric series. $\newline$ The sum $S_n$ of the first $n$ terms of a geometric series is given by the formula: $\newline$ $S_n = a \times (1 - r^n) / (1 - r)$ , where $a$ is the first term, $r$ is the common ratio, and $n$ is the number of terms. $\newline$ In this case, $a = 20$ , $r = 0.5$ , and $n = 6$ .
Calculate sum of terms: Calculate the sum of the first $6$ terms. $\newline$ $S_6 = 20 \times (1 - 0.5^6) / (1 - 0.5)$ $\newline$ $S_6 = 20 \times (1 - 0.015625) / 0.5$ $\newline$ $S_6 = 20 \times 0.984375 / 0.5$ $\newline$ $S_6 = 19.6875 / 0.5$ $\newline$ $S_6 = 39.375$ $\newline$ Since we need to round to the nearest integer, the sum is approximately $39$ .

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