Solving Т(n)=(1)/((1)/(T(n-1))+n^2) and T(1)=1 How do I go on from here? How can I find the solution? Thanks. :)

Quessyrutty6w 2022-07-23 Answered
Solving Т ( n ) = 1 1 T ( n 1 ) + n 2 and T ( 1 ) = 1
I did find T ( 2 ) = 1 1 + n 2 , but I don't know how to proceed.
How do I go on from here? How can I find the solution? Thanks. :)
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Answers (1)

Helena Howard
Answered 2022-07-24 Author has 12 answers
1 t n = 1 t n 1 + n 2
and the rest is smooth I think:
1 t 2 = 1 t 1 + 2 2
1 t 3 = 1 t 2 + 3 2 . . .
1 t n = 1 t n 1 + n 2 .
A summing of these equalities gives:
1 t n 1 t 1 = 2 2 + 3 2 + . . . + n 2
or
1 t n = 1 2 + 2 2 + 3 2 + . . . + n 2
or
1 t n = n ( n + 1 ) ( 2 n + 1 ) 6 .
The answer is:
t n = 6 n ( n + 1 ) ( 2 n + 1 )

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