Find the sum of:
1 1 1 1
----- + ----- + ----- + ... + ----------
√1+√2 √2+√3 √3+√4 √999+√1000
We observe that:
1 V(n+1) - Vn
--------------------- = --------------------- = V(n+1) - Vn ...... (#)
Vn + V(n+1) (n+1) - n
Therefore, substituting n=1,2,3,.....,997,998,999 in turn, the given expression
reduces to:
{(V2 - V1) + (V3 - V2) + (V4 - V3)+ ............+ (V998 - V997) + (V999 - V998) + (V1000 - V999)}
Obviously, all the real numbers except V1000 and V1 cancels out, leaving us with:
V1000 - V1
= 10*V10 -1
---------------
(The numerical value of the final answer is approximately equal to 30.6227766)
Edited on July 18, 2022, 7:51 am
Puzzle Solution
