Evaluate: (1 - 1/10)(1 - 1/11)(1 - 1/12)...(1 - 1/99)(1 - 1/100)
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calendarEducator since 2008
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Let us assume that the number is E.
==> E = (1-1/10) ( 1-1/11) ( 1- 1/12) .... (1-1/99)( 1- 1/100)
Let us rewrite the terms by using the common denominator.
==> 1- 1/10 = 10/10 - 1/10 = 9/10
==> 1- 1/11 = 11/11 - 1/11 = 10/11
==> 1- 1/12 = 12/12 - 1/12 = 11/12
.........
==> 1- 1/99 = 99/99 - 1/99 = 98/99
==> 1 - 1/100 = 100/100 - 1/100 = 99/100
Now we will rewrite the terms.
==> E = 9/10 * 10/11 * 11/12 ............ * 98/99 * 99/100
After we reduce similar terms.
==> E = 9 / 100
Then the number ( 1- 1/10)(1-1/11) ....(1-1/100) = 9/100
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calendarEducator since 2010
write12,544 answers
starTop subjects are Math, Science, and Business
We have to find (1 - 1/10)(1 - 1/11)(1 - 1/12)...(1 - 1/99)(1 - 1/100).
Now (1 - 1/10)(1 - 1/11)(1 - 1/12)...(1 - 1/99)(1 - 1/100).
=> [(10 - 1)/10][(11 - 1)/11][(12 - 1)/12]...[(99-1)/99][(100-1)/100].
=> (9/10)(10/11)(12/11)...(98/99)(99/100)
We see that all terms get cancelled except 9/100.
Therefore (1 - 1/10)(1 - 1/11)(1 - 1/12)...(1 - 1/99)(1 - 1/100) = 9/100.
Q: (1 - 1/10)(1 - 1/11)(1 - 1/12)...(1 - 1/99)(1 - 1/100).
A:
Let x = (1-1/10)(1-1/11)(1-1/12)...(1-1/100).
=> x = {(10-1)/10)}{(11-1)/11}{(12-1)/12}...{(100-1)/100}
=> x = (9/10)(10/11)(11/12)(12/13)....(99/100)
=> x= (9*10*11*12*...99)/(10*11*12*13*...100)
=> x = 9* {(10*11*12.13..99)/(10*11*12.13*...99)}/100
=> x = 9/100 = 0.09.
Therefore x = 0.09.
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