Show work and explain with formulas.23. Find the sum of the geometric series: 1/9 + 1/3 + 1 + ... + 218724. In a geometric sequence, the first term is 27 and the common ratio is 1/3. Find the sum of the first 6 terms.25. The sum of the first n terms of the geometric sequence is -6, -12, -24, ... is -762. Find the value of n.​

23 Answer:  [tex]\bold{\dfrac{29,524}{9}}[/tex]Step-by-step explanation:[tex]\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[/tex][tex]\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}}[/tex]24 Answer:  [tex]\bold{\dfrac{364}{9}}[/tex]Step-by-step explanation:[tex]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}}[/tex]25 Answer:  n=7Step-by-step explanation:[tex]\{-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}[/tex]