An algebra problem by Sabhrant Sachan

Algebra Level 3

1 + 4 1 + 5 1 + 6 1 + = ? \large \sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+\cdots }}}} = \, ?


The answer is 5.

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2 solutions

Sabhrant Sachan
May 4, 2016

we start with A=5 A = 25 A = 1 + 24 A = 1 + 4 × 6 A = 1 + 4 36 A = 1 + 4 1 + 5 × 7 A = 1 + 4 1 + 5 49 A = 1 + 4 1 + 5 1 + 6 × 8 A = 1 + 4 1 + 5 1 + 6 64 A = 1 + 4 1 + 5 1 + 6 1 + 7 Hence A = x = 5 \text {we start with A=5} \\ \implies A=\sqrt{25} \\ \implies A=\sqrt{1+24} \\ \implies A=\sqrt{1+4\times6} \\ \implies A=\sqrt{1+4\sqrt{36}} \\ \implies A=\sqrt{1+4\sqrt{1+5\times7}} \\ \implies A=\sqrt{1+4\sqrt{1+5\sqrt{49}}} \\ \implies A=\sqrt{1+4\sqrt{1+5\sqrt{1+6\times8}}} \\ \implies A=\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{64}}}} \\ \implies A=\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+7\cdots}}}} \\ \text{Hence} \boxed{A=x=5}

You make a "leap of faith" at the end, of course, as you casually move from the finite case to the limit ;)

Otto Bretscher - 5 years, 1 month ago

But this cannot be called as a solution to this problem...

Sarth Vitekar. - 4 years, 3 months ago

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Yeah because how do we know whether we have to start with 25 or any other number

Syed Hamza Khalid - 3 years, 9 months ago

Brilliant!

Apoorv Priyadarsh - 3 years, 8 months ago

This question is wrong as it can have infinite solutions...for example if we start with 3...with the same process we can have d result...but we have to continue with fractions...this infinite series was ramanujan's which can be started with any number...

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