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Question

Find the generating functions of the series 14+24x+34x2+44x3+54x4+....

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Solution

S=1+16x+81x2+256x3+625x4+1296x5

5xS=5x80x2405x31280x43125x5

10x2S=10x2+160x3+810x4+2560x5+

10x3S=10x3160x4810x5

5x4S=5x4+80x5+

x5S=x5

Adding the above series, we get

(13x+10x210x3+5x4x5)S=1+11x+11x2+x3(1x)5S=1+11x+11x2+x3S=1+11x+11x2+x3(1x)5

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