Mathematical sequence problems in grade one of senior high school In the sequence an BN, A1 = B1 = 1, an + 1 = an / 2An + 1, BN + 1-bn = 1 / an, Sn = 1 / B1 + 1 / B2 + 1 / B3 +. + 1 / BN 1. Find BN 2. It is proved that SN is less than nine fourths

Mathematical sequence problems in grade one of senior high school In the sequence an BN, A1 = B1 = 1, an + 1 = an / 2An + 1, BN + 1-bn = 1 / an, Sn = 1 / B1 + 1 / B2 + 1 / B3 +. + 1 / BN 1. Find BN 2. It is proved that SN is less than nine fourths

A (n + 1) = an / (2An + 1) a (n + 1) a (n + 1) a (n + 1) a (n + 1) a (n + 1) a (n + 1) a (n + 1) a (n + 1) a (n + 1) a (n + 1) a (n + 1) a (n + 1) a (n + 1) a (n +
1. Take the reciprocal of a (n + 1) = an / (2An + 1) to get 1 / a (n + 1) = 1 / an + 2
So the sequence {1 / an} is an arithmetic sequence, the first term is 1, and the tolerance is 2
So 1 / an = 1 / A1 + 2 (n-1) = 2N-1
Then B (n + 1) - BN = 2N-1
B（n）-B（n-1）=2n-3
.
B（2）-B（1）=1
Add the above equations to get B (n + 1) - B (1) = 1 + 3 + 5 + +2N-1 = n ^ 2
So B (n + 1) = n ^ 2 + B (1) = n ^ 2 + 1. Similarly, B (n) = (n-1) ^ 2 + 1 = n ^ 2-2n + 2
2. (this kind of topic can be proved by appropriate scaling)
Because n ^ 2-2n + 2 > n ^ 2-2n = n (n-2)
So 1 / (n ^ 2-2n + 2)