A monkey is swinging from a tree. On the first swing, she passes through an arc whose length is 24m. With each swing, she travels along an arc that is half as long as the arc of the previous swing. Which expression gives the total length the monkey swings in her first n swings? Choose 1 answer: (A) 24*((1-1.5^(n)))/(-0.5) (B) 24*((1-1.5^(n)))/(0.5) (c) 24*((1-0.5^(n)))/(-0.5) (D) 24*((1-0.5^(n)))/(0.5)

Identify the pattern: Identify the pattern of the lengths of the swings. The monkey swings through an arc of 24m on the first swing. Each subsequent swing is half the length of the previous swing. This is a geometric sequence where the first term a1 is 24m and the common ratio r is 0.5.Write the formula: Write the formula for the sum of the first n terms of a geometric sequence. The sum Sn of the first n terms of a geometric sequence is given by Sn = a1 * (1 - r^n) / (1 - r), where a1 is the first term and r is the common ratio.Substitute the values: Substitute the values of a1 and r into the formula. Here, a1 = 24m and r = 0.5. Substituting these values into the formula gives us Sn = 24 * (1 - 0.5^n) / (1 - 0.5).Simplify the denominator: Simplify the denominator. Since 1 - 0.5 equals 0.5, the formula simplifies to Sn = 24 * (1 - 0.5^n) / 0.5.Choose the correct answer: Choose the correct answer from the given options. The simplified formula matches option (D): 24 * ((1 - 0.5^n)) / (0.5).

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A monkey is swinging from a tree. On the first swing, she passes through an arc whose length is
24m. With each swing, she travels along an arc that is half as long as the arc of the previous swing.
Which expression gives the total length the monkey swings in her first
n swings?
Choose 1 answer:
(A)
24*((1-1.5^(n)))/(-0.5)
(B)
24*((1-1.5^(n)))/(0.5)
(c)
24*((1-0.5^(n)))/(-0.5)
(D)
24*((1-0.5^(n)))/(0.5)

A monkey is swinging from a tree. On the first swing, she passes through an arc whose length is $24 \mathrm{~m}$ . With each swing, she travels along an arc that is half as long as the arc of the previous swing. $\newline$ Which expression gives the total length the monkey swings in her first $n$ swings? $\newline$ Choose $1$ answer: $\newline$ (A) $24 \cdot \frac{\left(1-1.5^{n}\right)}{-0.5}$ $\newline$ (B) $24 \cdot \frac{\left(1-1.5^{n}\right)}{0.5}$ $\newline$ (C) $24 \cdot \frac{\left(1-0.5^{n}\right)}{-0.5}$ $\newline$ (D) $24 \cdot \frac{\left(1-0.5^{n}\right)}{0.5}$

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Math Problems

Algebra 1

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Full solution

Q. A monkey is swinging from a tree. On the first swing, she passes through an arc whose length is

24 \mathrm{~m}

. With each swing, she travels along an arc that is half as long as the arc of the previous swing.

\newline

Which expression gives the total length the monkey swings in her first

n

swings?

\newline

Choose

1

answer:

\newline

(A)

24 \cdot \frac{\left(1-1.5^{n}\right)}{-0.5}

\newline

(B)

24 \cdot \frac{\left(1-1.5^{n}\right)}{0.5}

\newline

(C)

24 \cdot \frac{\left(1-0.5^{n}\right)}{-0.5}

\newline

(D)

24 \cdot \frac{\left(1-0.5^{n}\right)}{0.5}

Identify the pattern: Identify the pattern of the lengths of the swings. $\newline$ The monkey swings through an arc of $24\,\text{m}$ on the first swing. Each subsequent swing is half the length of the previous swing. This is a geometric sequence where the first term $a_1$ is $24\,\text{m}$ and the common ratio $r$ is $0.5$ .
Write the formula: Write the formula for the sum of the first $n$ terms of a geometric sequence. $\newline$ The sum $S_n$ of the first $n$ terms of a geometric sequence is given by $S_n = a_1 \times (1 - r^n) / (1 - r)$ , where $a_1$ is the first term and $r$ is the common ratio.
Substitute the values: Substitute the values of $a_1$ and $r$ into the formula. $\newline$ Here, $a_1 = 24\,\text{m}$ and $r = 0.5$ . Substituting these values into the formula gives us $S_n = 24 \times (1 - 0.5^n) / (1 - 0.5)$ .
Simplify the denominator: Simplify the denominator. $\newline$ Since $1 - 0.5$ equals $0.5$ , the formula simplifies to $S_n = 24 \times (1 - 0.5^n) / 0.5$ .
Choose the correct answer: Choose the correct answer from the given options. $\newline$ The simplified formula matches option (D): $24 \times \left(\frac{1 - 0.5^n}{0.5}\right)$ .