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Find the sum of the first 9 terms of the following series, to the nearest integer.

54,36,24,dots
Answer:

Find the sum of the first $9$ terms of the following series, to the nearest integer. $\newline$ $54,36,24, \ldots$ $\newline$ Answer:

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Sequences: mixed review

Full solution

Q. Find the sum of the first

9

terms of the following series, to the nearest integer.

\newline

54,36,24, \ldots

\newline

Answer:

Identify pattern: Identify the pattern in the series. $\newline$ The series is $54, 36, 24, \ldots$ which seems to be a geometric series because each term is multiplied by a common ratio to get the next term. To find the common ratio $(r)$ , we divide the second term by the first term. $\newline$ $r = \frac{36}{54}$
Calculate common ratio: Calculate the common ratio. $\newline$ $r = \frac{36}{54} = \frac{2}{3}$ $\newline$ The common ratio is $\frac{2}{3}$ .
Use formula for sum: Use the formula for the sum of the first $n$ terms of a geometric series. $\newline$ The formula for the sum of the first $n$ terms ( $S_n$ ) of a geometric series is: $\newline$ $S_n = a \times (1 - r^n) / (1 - r)$ , where $a$ is the first term and $r$ is the common ratio. $\newline$ For this series, $a = 54$ , $r = 2/3$ , and $n = 9$ .
Substitute values and calculate: Substitute the values into the formula and calculate the sum. $\newline$ $S_9 = 54 \times (1 - (\frac{2}{3})^9) / (1 - \frac{2}{3})$
Evaluate expression: Evaluate the expression. $\newline$ $S_9 = 54 \times (1 - (\frac{2}{3})^9) / (\frac{1}{3})$ $\newline$ First, calculate $(\frac{2}{3})^9$ . $\newline$ $(\frac{2}{3})^9 = (\frac{2^9}{3^9})$
Calculate powers: Calculate $(2^9)$ and $(3^9)$ . $2^9 = 512$ $3^9 = 19683$ Now, substitute these values back into the expression.
Continue calculation: Continue the calculation. $\newline$ $S_9 = 54 \times (1 - \frac{512}{19683}) / (\frac{1}{3})$ $\newline$ $S_9 = 54 \times (\frac{19683}{19683} - \frac{512}{19683}) / (\frac{1}{3})$ $\newline$ $S_9 = 54 \times (\frac{19171}{19683}) / (\frac{1}{3})$
Simplify expression: Simplify the expression. $\newline$ $S_9 = 54 \times (19171/19683) \times 3$ $\newline$ $S_9 = 54 \times 3 \times (19171/19683)$ $\newline$ $S_9 = 162 \times (19171/19683)$
Perform division: Perform the division. $\newline$ $S_9 = 162 \times (19171/19683)$ $\newline$ $S_9 \approx 162 \times 0.974$
Multiply for sum: Multiply to find the sum. $\newline$ $S_9 \approx 162 \times 0.974$ $\newline$ $S_9 \approx 157.788$
Round to nearest integer: Round to the nearest integer. $S_9 \approx 158$

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