Simplify
√10+√1+√10+√2+⋯+√10+√99√10−√1+√10−√2+⋯+√10−√99
Scroll down for simplified form and solution.
The expression is equal to √2+1. Surprising!
Scroll down for solution.
Let a+b = 100.
Now
√10−√a√10+√a=√100−a=√b
Let u=√10+√a and v=√10−√a
So 20=u2+v2
and √b=uv
Thus
√20+2√b=√u2+v2+2uv=u+v
Similarly
√20−2√b=√u2+v2−2uv=u−v
Thus if
P=99∑a=1√10+√a
and
Q=99∑a=1√10−√a
We get
√2P=P+Q
and
√2Q=P−Q
This gives us PQ=√2+1
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