if 1/6+1/12+1/20+1/30.....+1/9900=?
Answers
answer of this is 59/100
Answer:49/100
Step-by-step explanation:
the question is 1/6+1/12+1/20+1/30+.....going on.......till.........+1/9900
if you experiment with it closely, then you can observe the following pattern-
1/6+1/12+1/20+1/30+.......+1/9900
=1 /2*3 + 1 /3*4 + 1 /4*5 + 1 /5*6 + ............+ 1 /99*100
this series is called telescopic series.
1/2*3 can be also written as 1/2-1/3, check that out!! So do the others!
therefore,
=1/2*3 + 1/3*4 + 1/4*5 + 1/5*6+...+1/99*100
=1/2-1/3+1/3-1/4+1/4-1/5+1/5-1/6+.........+1/99-1/100
If you pay keen attention you can figure out that a very magical cancellation happens here.
Look at the second term of the series, its -1/3. Now look at the next term, its +1/3.
Now what is -1/3+1/3? Aha! Its a big round egg!
so now, cut them out of this equation
look at the next terms, -1/4 and 1/4. Wow quite a number of cancellations!
So, we can cancel the numbers from the second to the second-last term.
Now what is left of this equation, its-
1/2 - 1/100
=50/100 - 1/100
=(50-1)/100
=49/100 (THATS IT!!!)
Now I have another question for you all,
1/2 + 1/6 + 1/12 + 1/20 + 1/30 +...................... till infinity
hint
take the last term and name it x*(x+1) and do it (infinity (+,-,*,/) x=infinity)and
infinity/infinity=1
you should get the answer as "1"
I hope my detailed and simplified explanation is clear for you all.