1.

Prove that (1 + x + x2 + x3 +….)  (1 + 3x + 6x2+ …)= (1 + 2x + 3x2 + …)2

Answer»

We know that

(1 – x)-1 = 1 + x + x2 + x3 +…           … (i)

and (1 – x)-3 = 1 + 3x + 6x2 + 10x3 + … (ii)

L.H.S. = (1 + x + x2 + x3 +…)(1 + 3x + 6x2 + …)

= {(1 – x)-1} {(1 – x)-3}

= {(1 – x)2}-2

= {(1 – x)-2}2

= (1 + 2x + 3x2 +…)2

= R.H.S.


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