Answer: Choice B) 1/3, 1/4, 1/5, 1/6
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Plug in n = 1 to find the first term
a_n = ( (n+1)! )/( (n+2)! )
a_1 = ( (1+1)! )/( (1+2)! )
a_1 = ( 2! )/( 3! )
a_1 = ( 2*1 )/( 3*2*1 )
a_1 = 2/6
a_1 = 1/3
The first term is 1/3. Optionally you can stop here because only choice B has 1/3 listed as the first term, so this must be the answer. However, I'm going to keep going to show how to find the three other terms. This will help confirm why choice B is the answer, and it will be handy for those times when you aren't given multiple choice answers.
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Plug in n = 2
a_n = ( (n+1)! )/( (n+2)! )
a_2 = ( (2+1)! )/( (2+2)! )
a_2 = ( 3! )/( 4! )
a_2 = ( 3*2*1 )/( 4*3*2*1 )
a_2 = 6/24
a_2 = 1/4
The second term is 1/4
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Plug in n = 3
a_n = ( (n+1)! )/( (n+2)! )
a_3 = ( (3+1)! )/( (3+2)! )
a_3 = ( 4! )/( 5! )
a_3 = ( 4*3*2*1 )/( 5*4*3*2*1 )
a_3 = 24/120
a_3 = 1/5
The third term is 1/5
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Plug in n = 4
a_n = ( (n+1)! )/( (n+2)! )
a_4 = ( (4+1)! )/( (4+2)! )
a_4 = ( 5! )/( 6! )
a_4 = ( 5*4*3*2*1 )/( 6*5*4*3*2*1 )
a_4 = 120/720
a_4 = 1/6
The fourth term is 1/6
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The first four terms are: 1/3, 1/4, 1/5, 1/6, so that confirms why choice B is the answer.
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