question about M.I

Question

Prove , by mathematical induction , that
 1x3+2x3^2+3x3^3+......+n(3^n)=3/4(3^n(2n-1)+1)for all positive integer n .

SP: You only need to do the part about......
Assume S(k) is true ...and below thanks

Audrey Hepburn · Accepted Answer

As follows~~~

                              
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heyhi · Answer

Assume S(k) is true,
i.e.  1x3+2x3^2+3x3^3+......+k(3^k)=3/4(3^k(2k-1)+1)

When n=k+1,
1x3+2x3^2+3x3^3+......+k(3^k) + (k+1) * 3^ (k+1)
= 3/4(3^k(2k-1)+1) + (k+1) * 3^ (k+1)
= 3/4[3^k(2k-1)+1 + (k+1) * 3^k * 3 * 4/3 ]
= 3/4[3^k(2k-1)+1 + 4(k+1) * 3^k  ]
= 3/4[3^k(2k-1+ 4(k+1)) + 1  ]
= 3/4(3^k(2(k+1)-1)+1)

so S(k+1) is also true

2007-10-18 21:45:41 補充：
sorry, some typing mistake for the last lineit should be ---= 3/4[3^k(6k 3)   1 ]= 3/4[3^(k 1)  (2k 1)   1 ]= 3/4[3^(k 1)  (2(k 1)-1) 1]

question about M.I

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