solve this 1/(1*2) + 1/(2*3) + 1/(3*4) + 1/(4*5) .....1/(999*1000)?

Answer 1

we have : 1/a - 1/(a+1) = (a + 1 - a)/(a * (a+1)) = 1/(a * (a+1))
so 1/(1*2) = 1 - 1/2
1/(2*3) = 1/2 - 1/3
1/(3*4) = 1/3 - 1/4
1/(4*5) = 1/4 - 1/5
................
1/(999*1000) = 1/999 - 1/1000
=> 1/(1*2) + 1/(2*3) + 1/(3*4) + 1/(4*5) .....1/(999*1000) = 1 - 1/1000 = 999/1000

Answer 2

1/1*2 = 1/2

1/1*2 + 1/2*3 = 1/2 + 1/6 = 2/3

1/1*2 + 1/2*3 + 1/3*4 = 2/3 + 1/12 = 3/4

1/1*2 + 1/2*3 + 1/3*4 +1/4*5 = 3/4 + 1/20 = 4/5

1/1*2 + 1/2*3 + 1/3*4 +1/4*5 + ... + 1/n*(n + 1) = n/(n + 1)

So 1/1*2 + 1/2*3 + 1/3*4 +1/4*5 + ... + 1/(999*1000 = 999/1000

Answer 3

a/a*(a+1) + a/(a+1)*(a+2) = 2/(a+2)
a/a*(a+1) + a/(a+1)*(a+2) + a/(a+2)*(a+3) = 3(a+3)
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thus
here a being '1'
if sereies go till adding 1/999*1000
it will be equal to 999/1000