23 Answer: \bold{\dfrac{29,524}{9}}
9
29,524
Explanation:
\begin{lgathered}\dfrac{1}{9}+\dfrac{1}{3}+1+...+2187\\\\\\a_1=\dfrac{1}{9}=3^{-2}\qquad r=3\qquad a_n=2187\\\\\underline{\text{Find n:}}\\a_n=a_1\cdot r^{n-1}\\2187=3^{-2}(3)^{n-1}\\2187=3^{n-3}\\3^7=3^{n-3}\\7=n-3\\10=n\end{lgathered}
9
1
+
3
1
+1+...+2187
a
1
=
9
1
=3
−2
r=3a
n
=2187
Find n:
a
n
=a
1
⋅r
n−1
2187=3
−2
(3)
n−1
2187=3
n−3
3
7
=3
n−3
7=n−3
10=n
\begin{lgathered}\underline{\text{Find the sum:}}\\S_n=\dfrac{a_1(1-r^n)}{1-r}\\\\\\S_{10}=\dfrac{\frac{1}{9}(1-3^{10})}{1-3}\\\\\\.\quad =\dfrac{1-59,049}{(9)(-2)}\\\\\\.\quad =\dfrac{-59,048}{9(-2)}\\\\\\.\quad =\large\boxed{\dfrac{29,524}{9}}\end{lgathered}
Find the sum:
S
n
=
1−r
a
1
(1−r
n
)
S
10
=
1−3
9
1
(1−3
10
)
.=
(9)(−2)
1−59,049
.=
9(−2)
−59,048
.=
9
29,524
24 Answer: \bold{\dfrac{364}{9}}
9
364
Explanation:
\begin{lgathered}a_1=27\qquad r=\dfrac{1}{3}\qquad n=6\\\\S_n=\dfrac{a_1(1-r^n)}{1-r}\\\\\\S_6=\dfrac{27(1-\frac{1}{3}^6)}{1-\frac{1}{3}}\\\\\\.\quad =\dfrac{27(\frac{728}{729})}{\frac{2}{3}}\\\\\\.\quad =\dfrac{27(728)}{729}\cdot \dfrac{3}{2}\\\\\\.\quad =\large\boxed{\dfrac{364}{9}}\end{lgathered}
a
1
=27r=
3
1
n=6
S
n
=
1−r
a
1
(1−r
n
)
S
6
=
1−
3
1
27(1−
3
1
6
)
.=
3
2
27(
729
728
)
.=
729
27(728)
⋅
2
3
.=
9
364
25 Answer: n=7
Explanation:
\begin{lgathered}\{-6,\ -12,\ -24,\ ...\ \}\\\\a_1=-6\qquad r=2\qquad S_n=-762\\\\S_n=\dfrac{a_1(1-r^n)}{1-r}\\\\\\-762=\dfrac{-6(1-2^n)}{1-2}\\\\\\-762=\dfrac{-6(1-2^n)}{-1}\\\\\\\dfrac{-762}{6}=1-2^n\\\\-127=1-2^n\\\\-128=-2^n\\\\128=2^n\\\\2^7=2^n\\\\\large\boxed{7=n}\end{lgathered}
{−6, −12, −24, ... }
a
1
=−6r=2S
n
=−762
S
n
=
1−r
a
1
(1−r
n
)
−762=
1−2
−6(1−2
n
)
−762=
−1
−6(1−2
n
)
6
−762
=1−2
n
−127=1−2
n
−128=−2
n
128=2
n
2
7
=2
n
7=n