Ramanujan (1887-1920) discovered some formulas on algebraic nested radicals. This article is based on one of those formulas. The main aim of this article is to discuss and derive them intuitively. Nested radicals have many applications in
Number Theory as well as in
Numerical Methods.
The simple
binomial theorem of degree 2 can be written as:
Replacing
by
where
, we can have
or,
Arranging terms in a way that
Taking Square-root of both sides
or,
Take a break. And now think about 
in the same way, as:
.
Therefore, in equation (2), if we replace 
by 
, we get
or, 
Similarly, 
and also, 
Similarly,
where, 
.
Putting the value of 
from equation (3) in equation (2), we get:
Again, putting the value of 
from equation (4) in equation (7), we get
Generalising the result for 
-nested radicals:
This is the general formula of Ramanujan Nested Radicals upto roots.
Some interesting points
As 
and all are real numbers, thus they can be interchanged with each other.
i.e.,
}^2+x\sqrt{a(x+n)+{(n+a)}^2+(x+n)\sqrt{a(x+2n)+{(n+a)}^2+(x+2n)\sqrt{\ldots+(x+(k-2)n)\\ \sqrt{a(x+(k-1)n)+{(n+a)}^2+x(x+(k+1)n+a)}}}}}) \\=\sqrt{an+{(x+a)}^2+n\sqrt{a(n+x)+{(x+a)}^2+(n+x)\sqrt{a(n+2x)+{(x+a)}^2+(n+2x)\sqrt{\ldots+(n+(k-2)x) \\ \sqrt{a(n+(k-1)x)+{(x+a)}^2+n(n+(k+1)x+a)}}}}}) \\=\sqrt{xa+{(n+x)}^2+a\sqrt{x(a+n)+{(n+x)}^2+(a+n)\sqrt{x(a+2n)+{(n+x)}^2+(a+2n)\sqrt{\ldots+(a+(k-2)n) \\ \sqrt{ x(a+(k-1)n)+{(n+x)}^2+a(a+(k+1)n+x)}}}}} \ \ldots (10) "\tiny \dpi{150} x+n+a = \\ \sqrt{ax+{(n+a)}^2+x\sqrt{a(x+n)+{(n+a)}^2+(x+n)\sqrt{a(x+2n)+{(n+a)}^2+(x+2n)\sqrt{\ldots+(x+(k-2)n)\\ \sqrt{a(x+(k-1)n)+{(n+a)}^2+x(x+(k+1)n+a)}}}}}) \\=\sqrt{an+{(x+a)}^2+n\sqrt{a(n+x)+{(x+a)}^2+(n+x)\sqrt{a(n+2x)+{(x+a)}^2+(n+2x)\sqrt{\ldots+(n+(k-2)x) \\ \sqrt{a(n+(k-1)x)+{(x+a)}^2+n(n+(k+1)x+a)}}}}}) \\=\sqrt{xa+{(n+x)}^2+a\sqrt{x(a+n)+{(n+x)}^2+(a+n)\sqrt{x(a+2n)+{(n+x)}^2+(a+2n)\sqrt{\ldots+(a+(k-2)n) \\ \sqrt{ x(a+(k-1)n)+{(n+x)}^2+a(a+(k+1)n+x)}}}}} \ \ldots (10)")
etc.
Putting 
in equation (9)
we have
or just, 
Again putting 
in (9)
Putting in equation (8)
Again putting 
=n(say) then
or, 
Putting 
in (15)
Putting 
and 
in (9) we get even numbers
Similary putting and 
in (9) we get a formula for odd numbers:
}^2+n\sqrt{2n+{(n+1)}^2+2n\sqrt{3n+{(n+1)}^2+3n\sqrt{\ldots+(k-1)n\sqrt{kn+{(n+1)}^2+(k+2)n^2+n}}}}} \ \ldots (18) "\tiny \dpi{150} 2n+1 =\sqrt{n+{(n+1)}^2+n\sqrt{2n+{(n+1)}^2+2n\sqrt{3n+{(n+1)}^2+3n\sqrt{\ldots+(k-1)n\sqrt{kn+{(n+1)}^2+(k+2)n^2+n}}}}} \ \ldots (18)")
or,
}^2+n\sqrt{2n+{(n+1)}^2+2n\sqrt{3n+{(n+1)}^2+3n\sqrt{\ldots+(k-1)n\sqrt{(k+3)n^2+(k+3)n+1}}}}} \ \ldots (19) "\tiny \dpi{150} 2n+1 =\sqrt{n+{(n+1)}^2+n\sqrt{2n+{(n+1)}^2+2n\sqrt{3n+{(n+1)}^2+3n\sqrt{\ldots+(k-1)n\sqrt{(k+3)n^2+(k+3)n+1}}}}} \ \ldots (19)")
Comments?
26.740278
83.888889
Like this:
Like Loading...
[...] are the pics for raw work done for Post On Ramanujan’s Nested Radicals #gallery-1 { margin: auto; } #gallery-1 .gallery-item { float: left; margin-top: 10px; text-align: [...]
wow! i’m impressed!…and you’re reading…my blog?
uh…i’m flattered…hope 2012 brings you more math puzzles to solve…
this is very helpful to me and you are doing a great job . thank you
hi..i am utkarsh.i have been working on a formula and i am stuck in nested radicals.
basically, i want to find out value of sqrt(2+sqrt(2+sqrt(2…………….sqrt(2)
for x of times,for example, for x=3, i want value of sqrt(2+sqrt(2+sqrt(2+sqrt(2))))
would you please help me?
by the way,are you left-handed,your hand writing is similiar to mine!
Dear Utkarsh! Thanks for reading the post. Before I comment, I would like to mention that Ramanujan Nested Radical formulas are proposed for infinte number of radicals in a number. When, there are finite number of nested radicals, the exact numerical value is calculated by an advanced calculator.
always means 
or approximately 1.4142… Similarly 
has its own numerical value. And so on. As we increases the number of squareroots, the value tends to 2 (not exactly 2).
upto infinte terms
.
Let me be clear.
But when infinite terms are considered, the numerical values cam be easily calculated using algebraic equations.
Let
or,
The non-negative solution of above quadratic equation is the numerical value of the nested radical (i.e., N=2).
What is ramanujan redical ?
thanks for the answer!i guess i will really have to use calculators!