Root 30

Algebra Level 2

Evaluate:

30 + 30 + 30 + 30 + 30 + 30 + . . . \sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { ... } } } } } } }


The answer is 6.

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2 solutions

Indronil Ghosh
Apr 26, 2014

Let 30 + 30 + 30 + = x \sqrt{30+\sqrt{30+\sqrt{30+\sqrt{\cdots}}}}=x .

Then, x = 30 + x x=\sqrt{30+x} .

So, x 2 x 30 = 0 x^2-x-30=0 , which can be factored into ( x 6 ) ( x + 5 ) = 0 (x-6)(x+5)=0 .

We take the quadratic's positive root to be our answer (since we want the principal square root), so x = 6 x=\boxed{6} .

Did it the same way... :)

Shreya R - 7 years ago

Great answer dude

Claudemir Silva - 6 years, 11 months ago

suppose let it be a terminating sequence answer will be sqrt(30n).n is the number of sqaureroots or thirties.....the answer will tend towards six for a non terminating sequence.

Vignesh Rajendran - 6 years, 4 months ago
Gandoff Tan
Apr 15, 2019

30 + 30 + 30 + 30 + 30 + 30 + = l e t x 30 + x = x 30 + x = x 2 x 2 x 30 = 0 ( x 5 ) ( x + 6 ) = 0 x = 5 ( x > 0 ) 30 + 30 + 30 + 30 + 30 + 30 + = 5 \begin{aligned} \sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { \cdots } } } } } } } & \overset { let }{ = } & x \\ \sqrt { 30+x } & = & x \\ 30+x & = & { x }^{ 2 } \\ { x }^{ 2 }-x-30 & = & 0 \\ (x-5)(x+6) & = & 0 \\ x & = & 5\quad (x>0) \\ \sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { 30+\sqrt { \cdots } } } } } } } & = & \boxed { 5 } \end{aligned}

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