Write first four terms of the following sequences. If the sequence appears to converge, make a conje

Finley Mckinney

Finley Mckinney

Answered question

2022-06-13

Write first four terms of the following sequences. If the sequence appears to converge, make a conjecture about its limit
a n + 1 = 1 + a n 2 ; a 0 = 2

Answer & Explanation

Arcatuert3u

Arcatuert3u

Beginner2022-06-14Added 30 answers

Given: a n + 1 = 1 + a n 2 ; a 0 = 2

Put n = 0
a 0 + 1 = 1 + a 0 2
= 1 + 2 2
= 2
a 1 = 2

Put n = 1
a 1 + 1 = 1 + a 1 2
= 1 + 2 2
= 2
a 2 = 2

Put n = 2
a 2 + 1 = 1 + a 2 2
= 1 + 2 2
= 2
a 3 = 2

Put n = 3
a 3 + 1 = 1 + a 3 2
= 1 + 2 2
= 2
a 4 = 2

Thus, we get the sequence 2 , 2 , 2 , 2 , . . .
This is a constant sequence and it’s convergent
For the limit:
lim n a n + 1 = lim n ( 1 + a 2 2 )
L = 1 + L 2
L = 2
Thus, the limit is 2

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