I am a city in northern India
known for my well planned architecture.
Gardens of rock and bangles I have,
valleys to relax, and lakes to unwind.
A union territory, and a capital of  two,
click on me, and find out who I am.

I lie on the eastern arm
I belong to a Himalayan town.
English men have grown their tea
public schools can be found in plenty.
There was a movement to set me free,
it ended in the formation of a GTA committee.
I am a beautiful town, not a city
Click here to find out more about me.

I sit on the edge of the peninsula
very close to the southern tip.
I was not a part of India until '63
the French weren't willing to set me free.
Spirits are high in people with me
Aurobindo himself found it hard to leave me.
I have a lot of peace within
find me and step in for a wiser thee.

The Thar desert is centered around me
sand dunes, desert towns, bush lands you will see.
Dry, dead, and barren I might be,
I'm still called the Golden City.
Camels to ride, and chai to drink,
shopping to be done, and heritages to visit.
Many have come and created memories,
look at me, and you'll be tempted to plan a see.

Ramanujan Tests

Let sq(x) imply the positive square root of a positive number.
Consider a sequence of functions as follows:-

f₁(x) = sq(1+sq(x))
f₂(x) = sq(1+sq(1+2*sq(x)))
f₃(x) = sq(1+sq(1+2*sq(1+3*sq(x))))
.
.
f_n(x) = sq(1+sq(1+2*sq(1+3*sq(... (1+n*sq(x))..)))

Evaluate this function as n tends to infinity.

Find solution!