A pendulum's initial swing is 80 cm, 76cm on its second swing, 72.2cm on its third swing, and 68.59cm on its fourth swing. Complete parts a and b below.&nbsp;a. What precise formula can be applied to determine the length of the n-th swing if the pattern persists?an=80(0.95)n−1&nbsp;(Simplify your answer. Use integers or decimals for any numbers in the expression.)&nbsp;b. Use your formula to find the distance of the 10-th swing.&nbsp;a10=?cm&nbsp;(Type an integer or decimal rounded to two decimal places as needed.)

Question

A pendulum&#039;s initial swing is 80 cm, 76cm on its second swing, 72.2cm on its third swing, and 68.59cm on its fourth swing. Complete parts a and b below.&amp;nbsp;a. What precise formula can be applied to determine the length of the n-th swing if the pattern persists?an=80(0.95)n−1&amp;nbsp;(Simplify your answer. Use integers or decimals for any numbers in the expression.)&amp;nbsp;b. Use your formula to find the distance of the 10-th swing.&amp;nbsp;a10=?cm&amp;nbsp;(Type an integer or decimal rounded to two decimal places as needed.)

faldduE · Accepted Answer

Step 1  Given, a pendulum swings 80 cm on its first swing, 76 on its second swing, and 72.2 on its third swing, and 68.69 cm on its fourth swing.  Step 2  Part (a),  A pendulum swings as follows:  First swing, a1=80cm  Second swing, a2=76cm  Third swing, a3=72.2cm  Fourth swing, a4=68.59cm  Common ratio, r1=a2a1=7680=0.95  r2=a3a2=72.276=0.95  Since the common ratio is same, so the pendulum swings are in geometric sequence.  First term, a=80  Common ratio, r=0.95  The n-th term of the geometric sequence is given by  ∴an=a(r)n−1; where n= number of terms  ⇒an=80(0.95)n−1  The distance of n-th swings is an=80(0.95)n−1.  Step 3  Part(b),  ∴an=80(0.95)n−1  Out n=10, then  ⇒a10=80(0.95)10−1  ⇒a10=80(0.95)9  ⇒a10=50.4199  ⇒a10=50.42cm  The distance of the 10-th swing is a10=50.42cm

A pendulum swings 80cm on its first swing, 76cm on its second swing, 72.2cm on i

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