Q. 63.7( 3 Votes )

Sum the following series to n terms :

1 + 4 + 13 + 40 + 121 + ……….

Answer :

Let s=1 + 4 + 13 + 40 + 121 + …………. + n


By shifting each term by one


S = 1 + 4 + 13 + 40 + 121 + …………. + nth ….(1)


S = 1 + 4 + 13 + 40 + …………. + (n - 1)th + nth …(2)


by (1) - (2) we get


0 = 1 + (3 + 9 + 27 + 81 + …….nth - (n - 1)th - n)


Nth = 1 + (3 + 32 + 33 + 34 + …….nth - (n - 1)th - nth)


Nth = 1 + (3 + 32 + 33 + …….3n - 1) ………(3)


we know



Substituting the above-given value in (3)




thus


s = 1 + 4 + 13 + 40 + 121 + …………. +


We know by property that:


∑axn + bxn - 1 + cxn - 2…….d0=a∑xn + b∑xn - 1 + c∑xn - 2…….. + d0∑1


………….. (4)


We know





Thus substituting the above values in(4)




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