Question 13 & 14 Exercise 4.5
Solutions of Question 13 & 14 of Exercise 4.5 of Unit 04: Sequence and Series. This is unit of A Textbook of Mathematics for Grade XI is published by Khyber Pakhtunkhwa Textbook Board (KPTB or KPTBB) Peshawar, Pakistan.
Question 13
If y=x3+x232+x333+… where 0<x<3, then show that x=3y1+y.
Solution
Adding 1 to both sides of the given series, we get 1+y=1+x3+x232+x333 Now the series on R.H.S of the above is geometric series with a1=1,
r=x3, and |r|=x3<1 because 0<x<3.
Thus infinite sum exists and is given by S∞=a11−r, putting a1,r, we get S∞=11−x3=33−x putting in (i), we have 1+y=33−x⇒3−x=31+y⇒x=3−31+y⇒x=3+3y−31+y⇒x=3y1+y which is required result.
Question 14
A ball rebound to half the height from which it is dropped. If it is dropped from 10ft, how far does it travel from the moment it dropped until the moment of its eighth bounce?
Solution
S=10+[10(12)+10(12)]+[10(12)2+10(12)2]+…+[10(12)7+10(12)7]S=10+2[10(12)+10(12)2+10(12)7]...(i)
The sequence is bracket is geometric sequence,
with a1=10(12),r=12,n=7.
Then S7=10(12)[1−(12)7]1−12
⇒S7=10[1−127]⇒S7=10,0.9921875=9.921875
Putting (ii) in (i), we get
S=10+2(9.921875)=10+19.84375S=29.84375 feet approximatelyS=292732ft
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