How do you find the number of terms given #s_n=3829# and #7+(-21)+63+(-189)+...#?

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Answer 1 · 2016-08-21T21:57:28

#n = 6#

#7+(-21)+63+(-189)+cdots = 7sum_{k=0}^{k=n}(-3)^k = S_n#

We know that

#sum_{k=0}^{k=n}z^k = (z^{n+1}-1)/(z-1)#

so

#S_n = 7((-3)^{n+1}-1)/(-3-1) = 3829#

then

#(-3)^{n+1}=(-4)3829/7+1#

and

#(-3)^n = (4 cdot 3829/7-1)/3 = 729 = 3^6#

so

#n = 6#