數學知識交流---PolyU Math 2~3Jan12

1.
Solve:
a + 1/a = 2
(a + 1/a)³ = 8
a³ + 3(a + 1/a) + 1/a³ = 8
a³ + 3 × 2 + 1/a³ = 8
a³ + 1/a³ = 2

a + 1/a = 2
(a + 1/a)⁵ = 32
a⁵ + 1/a⁵ + 5(a³ + 1/a³) + 10(a + 1/a) = 32
a⁵ + 1/a⁵ + 5 × 2 + 10 × 2 = 32
a⁵ + 1/a⁵ = 2

2.
Solve:
s = 1 - 1/3 + 1/8 - 1/15 + 1/24 - 1/35 + 1/48 - ...
= 1 + [1/(2×4) + 1/(4×6) + 1/(6×8) + ...] - [1/(1×3) + 1/(3×5) + 1/(5×7) + ...]
= 1 + (1/2 - 1/4 + 1/4 - 1/6 + ... )/2 - (1 - 1/3 + 1/3 - 1/5 + ...)/2
= 1 + 1/4 - 1/2
= 3/4
so 4s = 3.