Question

Evaluate:


`\bf{\sum^(20)_1\:4(\cfrac{8}{9})^(n-1)`



(find the sum of the first 20 terms of the geometric series)



Please help A.S.A.P. & show work as well.


Thank you guys!



`\bigstar`

Answer 1

Answer:
`\text \large S_(20) \ = \sf (4\left(9^(20)-8^(20)\right))/(3^(38)) \ \ \ or \ \ \ 32.586`

Given expression:

`\boxed{\sf \sum _(n=1)^(20)\:4\left((8)/(9)\right)^(n-1)}`

Identify the following:

• First Term (a) = 4(8/9)¹⁻¹ = 4

• Common ratio (r) = 8/9

• Total Terms (n) = 20

Formula Required:

`\rightarrow \quad \sf S_n = (a(r^n - 1))/(r-1)`

Insert values identified:

`\rightarrow \sf S_(20) = (4((8)/(9)^(20) - 1))/((8)/(9) -1) \quad\overset{simplify}{\longrightarrow} \quad (4\left(9^(20)-8^(20)\right))/(3^(38)) \quad \xrightarrow{\text{In \ Decimals} }\quad 32.58609013`