How do you find the number of terms given $s_{n} = - 66.67$ $s_n=-66.67$ and $-90+30+(-10)+10/3+...$ ?

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Answer 1 · 2016-08-15T14:34:57

The sum of first $4$ terms is $s_4=-66.67$

Let us denote, by $t_n$ the $n^(th)$ term of the given series.Then, we find
that, $t_2/t_1=t_3/t_2=t_4/t_3=...=-1/3$ .

We conclude that it is a Geometric Series, with, common ratio

$r=-1/3, and, t_1=-90$

The sum $s_n$ of first $n$ terms of the series is given by,

$s_n=(t_1(1-r^n))/(1-r)$

Our goal is to find $n$ , given, $s_n=-66.67=-66 2/3=-200/3, &, r=-1/3$ .

$:. -200/3=(-90){(1-(-1/3)^n)/(1+1/3)}$

$:. -200/3(1/-90)(4/3)=80/81={1-(-1/3)^n}$

$:. (-1/3)^n=1-80/81=1/81=(-1/3)^4$

$:. n=4$

Hence, the sum of first $4$ terms is $s_4=-66.67$

How do you find the number of terms given s_n=-66.67sn=−66.67s_n=-66.67 and -90+30+(-10)+10/3+...-90+30+(-10)+10/3+...?