Question

a=24 r=1/2
Calculate the total distance the monkey has travelled from 5th swing to 25th swing.

Answer 1

$1.4976$

Step-by-step explanation:

$we \: \: know \: \: that \\ \boxed{s_{n} = \frac{a(1 - r^{n} )}{1 - r} } \\ given \: \: a = 24 \: \: and \: \: r = \frac{1}{2} \\A/Q, \: \: find \: \: s_{25} - s_{5} = ? \\ putting \: \: the \: \: values \\ s_{25} - s_{5} = \: \frac{24 \times (0.0312)}{ \frac{1}{2} } \\ = 2 \times 24 \times 0.0312 = 1.4976$

Answer 2

Given : Distance covered by monkey in each swing is in  Geometric sequence

a= 24

r = 1/2

To Find :

total distance the monkey has travelled from 5th swing to 25th swing.

Solution:

Geometric sequence

A sequence of numbers in which the ratio between consecutive terms is constant and called the common ratio.

a , ar , ar² , ... , arⁿ⁻¹

The nth term of a geometric sequence with the first term a and the common ratio r is given by:   aₙ = arⁿ⁻¹

Sum is given by  Sₙ = a(rⁿ - 1)/(r - 1)

Sum of  infinite series is given by  a/(1 - r)   where -1 < r < 1  

a = 24

r = 1/2

distance the monkey has travelled from 5th swing to 25th swing

= Distance  travelled in 25 swings - Distance  travelled in 4 swings

=  24( (1/2)²⁵ - 1)/(1/2 - 1)     - 24( (1/2)⁴ - 1)/(1/2 - 1)

= -48 ( (1/2)²⁵ -  (1/2)⁴ )

≈  2.99999

≈ 3

Hence Distance covered  from 5th swing to 25th swing is about 3 unit

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