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\[a_n=a_{n-1}+1/a_{n-1}\] Where, n>1 \[ a_1=1\] Prove that: \[12 < a_{75} < 15\]

Note by Kïñshük Sïñgh 2 years, 3 months ago

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Here we go, \[a_{n}=a_{n-1}+1/a_{n-1}\]

Squaring both sides: \[a_{n}^{2}=a_{n-1}^{2}+1/a_{n-1}^{2}+2\]

So, if we remove the term: \[1/a_{n-1}^{2}\] from RHS

Then, we can say that: \[a_{n}^{2}>a_{n-1}^{2}+2\]

Therefore, applying a=1,2,3..till \[a_{75}\] and \[a_{1} =1\] \[a_{2}^{2}>3\] \[a_{3}^{2}>5\] .... .... ... Series is \[2n-1\]

Therefore, \[a_{75}^{2}>149\] \[a_{75}>12\]

Now, lets do another part: \[a_{n}^{2}=a_{n-1}^{2}+1/a_{n-1}^{2}+2\]

As, we know that : Maximum value of \[1/a_{n-1}^{2}=1\] Therefore, if we remove this and add 1 then, we can say that: \[a_{n}^{2}≤a_{n-1}^{2}+3\]

Now, \[a_{2}^{2}≤4\] \[a_{3}^{2}<7\] \[a_{4}^{2}<10\] ... .... ..... Series is 3n-2 type:

Therefore, \[a_{75}^{2}<223\] \[a_{75}<15\]

Hence, #\[12<a_{75}<15\] – Kïñshük Sïñgh · 2 years, 3 months ago

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@Kïñshük Sïñgh – Good solution – Ronak Agarwal · 2 years, 3 months ago

@Ronak Agarwal – Thanks :) – Kïñshük Sïñgh · 2 years, 3 months ago

Is this question asked by your sir or what.This question was also asked by my friends in school. – Ronak Agarwal · 2 years, 3 months ago

@Ronak Agarwal – I tried this... But finally i got so many equations... Which were very complex – Kïñshük Sïñgh · 2 years, 3 months ago

@Ronak Agarwal – No, i go through this question while surfing on Internet – Kïñshük Sïñgh · 2 years, 3 months ago

This question's solution can be found here – Dinesh Chavan · 2 years, 3 months ago

@Dinesh Chavan – Sorry but.... Solution is not there.. Check it again plz – Kïñshük Sïñgh · 2 years, 3 months ago

@Kïñshük Sïñgh – Dont worry, I will try to add a solution there – Dinesh Chavan · 2 years, 3 months ago

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## Comments

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TopNewestHere we go, \[a_{n}=a_{n-1}+1/a_{n-1}\]

Squaring both sides: \[a_{n}^{2}=a_{n-1}^{2}+1/a_{n-1}^{2}+2\]

So, if we remove the term: \[1/a_{n-1}^{2}\] from RHS

Then, we can say that: \[a_{n}^{2}>a_{n-1}^{2}+2\]

Therefore, applying a=1,2,3..till \[a_{75}\] and \[a_{1} =1\] \[a_{2}^{2}>3\] \[a_{3}^{2}>5\] .... .... ... Series is \[2n-1\]

Therefore, \[a_{75}^{2}>149\] \[a_{75}>12\]

Now, lets do another part: \[a_{n}^{2}=a_{n-1}^{2}+1/a_{n-1}^{2}+2\]

As, we know that : Maximum value of \[1/a_{n-1}^{2}=1\] Therefore, if we remove this and add 1 then, we can say that: \[a_{n}^{2}≤a_{n-1}^{2}+3\]

## Note that equals to sign will be for n=2 only

Now, \[a_{2}^{2}≤4\] \[a_{3}^{2}<7\] \[a_{4}^{2}<10\] ... .... ..... Series is 3n-2 type:

Therefore, \[a_{75}^{2}<223\] \[a_{75}<15\]

Hence, #\[12<a_{75}<15\] – Kïñshük Sïñgh · 2 years, 3 months ago

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– Ronak Agarwal · 2 years, 3 months ago

Good solutionLog in to reply

– Kïñshük Sïñgh · 2 years, 3 months ago

Thanks :)Log in to reply

Is this question asked by your sir or what.This question was also asked by my friends in school. – Ronak Agarwal · 2 years, 3 months ago

Log in to reply

– Kïñshük Sïñgh · 2 years, 3 months ago

I tried this... But finally i got so many equations... Which were very complexLog in to reply

– Kïñshük Sïñgh · 2 years, 3 months ago

No, i go through this question while surfing on InternetLog in to reply

This question's solution can be found here – Dinesh Chavan · 2 years, 3 months ago

Log in to reply

– Kïñshük Sïñgh · 2 years, 3 months ago

Sorry but.... Solution is not there.. Check it again plzLog in to reply

– Dinesh Chavan · 2 years, 3 months ago

Dont worry, I will try to add a solution thereLog in to reply