am mi

CE level's solution:

Let the preposition P(n) be "1/2x5 + 1/5x8 +1/8x11+................+1/(3n-1)(3n+2)=n/6n+4" where n is an positive integer.

When n=1, L.H.S. = 1/10 = 1 / 6(1) + 4 = R.H.S.
P(1) is true.

Assume P(k) is true for some positive integer k,
i.e. 1/2x5 + 1/5x8 +1/8x11+................+1/(3k-1)(3k+2)=k/6k+4

When n = k+1,
L.H.S.
= 1/2x5 + 1/5x8 +1/8x11+................+1/(3k-1)(3k+2) + 1/(3k+2)(3k+5)
= k/6k+4 + 1/(3k+2)(3k+5) ( by assumption )
= [ k ( 3k+5 ) + 2 ] / (6k+4)(3k+5)
= ( k + 1 )( 3k + 2 ) / ( 6k+4 )( 3k+5 )
= ( k + 1 ) / ( 6k + 10 )
= ( k+1 ) / [ 6(k+1) + 4 ]
P( k+1) is true then.

By Mathematic Induction, P(n) is true for all positive integer n.

好詳細~~