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Question Number 206038 by BaliramKumar last updated on 05/Apr/24
(√(1 + 2023(√(1 + 2024(√(1+ 2025(√(1 + 2026(√(1 + ..............∞)))))))))) = ?
$$\sqrt{\mathrm{1}\:+\:\mathrm{2023}\sqrt{\mathrm{1}\:+\:\mathrm{2024}\sqrt{\mathrm{1}+\:\mathrm{2025}\sqrt{\mathrm{1}\:+\:\mathrm{2026}\sqrt{\mathrm{1}\:+\:…………..\infty}}}}}\:=\:?\:\:\:\:\:\:\:\:\:\:\:\:\:\:\:\: \\ $$
Answered by mr W last updated on 05/Apr/24
x+1=(√(1+x(√(1+(x+1)(√(1+(x+2)(√(1+...))))))))  ⇒2024=(√(1 + 2023(√(1 + 2024(√(1+ 2025(√(1 + 2026(√(1 + ..............∞))))))))))
$${x}+\mathrm{1}=\sqrt{\mathrm{1}+{x}\sqrt{\mathrm{1}+\left({x}+\mathrm{1}\right)\sqrt{\mathrm{1}+\left({x}+\mathrm{2}\right)\sqrt{\mathrm{1}+…}}}} \\ $$$$\Rightarrow\mathrm{2024}=\sqrt{\mathrm{1}\:+\:\mathrm{2023}\sqrt{\mathrm{1}\:+\:\mathrm{2024}\sqrt{\mathrm{1}+\:\mathrm{2025}\sqrt{\mathrm{1}\:+\:\mathrm{2026}\sqrt{\mathrm{1}\:+\:…………..\infty}}}}} \\ $$
Answered by MATHEMATICSAM last updated on 05/Apr/24
This problem is based on Ramanujan′s  infinite nested redical problem.  (n + 1)^2  = n^2  + 2n + 1  ⇒ n + 1 = (√(1 + n(n + 2)))  If we put 2023 as the value of n then it will be  2024 = (√(1 + (2023)(2025)))  If we put 2024 as the value of n then it will be  2025 = (√(1 + (2024)(2026)))  So we can say  2026 = (√(1 + (2025)(2027)))  Now 2025 = (√(1 + 2024(√(1 + (2025)(2027)))))  Now 2024 = (√(1 + 2023(√(1 + 2024(√(1 + 2025(√(...))...))))))  Thus this will continue to infinity  2027 = (√(1 + (2026)(2028)))  So 2024 is the answer.
$$\boldsymbol{\mathrm{This}}\:\boldsymbol{\mathrm{problem}}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{based}}\:\boldsymbol{\mathrm{on}}\:\boldsymbol{\mathrm{Ramanujan}}'\boldsymbol{\mathrm{s}} \\ $$$$\boldsymbol{\mathrm{infinite}}\:\boldsymbol{\mathrm{nested}}\:\boldsymbol{\mathrm{redical}}\:\boldsymbol{\mathrm{problem}}. \\ $$$$\left({n}\:+\:\mathrm{1}\right)^{\mathrm{2}} \:=\:{n}^{\mathrm{2}} \:+\:\mathrm{2}{n}\:+\:\mathrm{1} \\ $$$$\Rightarrow\:{n}\:+\:\mathrm{1}\:=\:\sqrt{\mathrm{1}\:+\:{n}\left({n}\:+\:\mathrm{2}\right)} \\ $$$$\mathrm{If}\:\mathrm{we}\:\mathrm{put}\:\mathrm{2023}\:\mathrm{as}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{n}\:\mathrm{then}\:\mathrm{it}\:\mathrm{will}\:\mathrm{be} \\ $$$$\mathrm{2024}\:=\:\sqrt{\mathrm{1}\:+\:\left(\mathrm{2023}\right)\left(\mathrm{2025}\right)} \\ $$$$\mathrm{If}\:\mathrm{we}\:\mathrm{put}\:\mathrm{2024}\:\mathrm{as}\:\mathrm{the}\:\mathrm{value}\:\mathrm{of}\:{n}\:\mathrm{then}\:\mathrm{it}\:\mathrm{will}\:\mathrm{be} \\ $$$$\mathrm{2025}\:=\:\sqrt{\mathrm{1}\:+\:\left(\mathrm{2024}\right)\left(\mathrm{2026}\right)} \\ $$$$\mathrm{So}\:\mathrm{we}\:\mathrm{can}\:\mathrm{say} \\ $$$$\mathrm{2026}\:=\:\sqrt{\mathrm{1}\:+\:\left(\mathrm{2025}\right)\left(\mathrm{2027}\right)} \\ $$$$\mathrm{Now}\:\mathrm{2025}\:=\:\sqrt{\mathrm{1}\:+\:\mathrm{2024}\sqrt{\mathrm{1}\:+\:\left(\mathrm{2025}\right)\left(\mathrm{2027}\right)}} \\ $$$$\mathrm{Now}\:\mathrm{2024}\:=\:\sqrt{\mathrm{1}\:+\:\mathrm{2023}\sqrt{\mathrm{1}\:+\:\mathrm{2024}\sqrt{\mathrm{1}\:+\:\mathrm{2025}\sqrt{…}…}}} \\ $$$$\mathrm{Thus}\:\mathrm{this}\:\mathrm{will}\:\mathrm{continue}\:\mathrm{to}\:\mathrm{infinity} \\ $$$$\mathrm{2027}\:=\:\sqrt{\mathrm{1}\:+\:\left(\mathrm{2026}\right)\left(\mathrm{2028}\right)} \\ $$$$\boldsymbol{\mathrm{So}}\:\mathrm{2024}\:\boldsymbol{\mathrm{is}}\:\boldsymbol{\mathrm{the}}\:\boldsymbol{\mathrm{answer}}. \\ $$
Commented by TonyCWX08 last updated on 13/Apr/24
Radical
$${Radical} \\ $$

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