Question

Find the sum of the following series. Round to the nearest hundredth if necessary 5+10+20+...+163840

Answer 1

1. The first step is to identify this as a geometric series.

Remember you can calculate the sum of a geometric series using the formula:

`S_n=(a(r^n-1))/(r-1)`

Where:

• S, is the sum

,

• a, is the first term

,

• r ,is the common ratio (what you're multiplying by to get to the next term)

,

• n ,is the number of terms you're adding up

In this case, we have to calculate wich n corresponds to 163840, using the geometric sequence:

`T_n=ar^(n-1)`

This way,

`\begin{gathered} 163840=(5)(2^(n-1))\rightarrow163840=(5)(2^n)(2^(-1)) \\ \rightarrow(163840)/(2^(-1))=(5)(2^n)\rightarrow327680=(5)(2^n) \\ \rightarrow(327680)/(5)=(2^n)\rightarrow2^n=65536 \\ \rightarrow n=\log _2(65536)\rightarrow n=16 \end{gathered}`

Now, let's use the formula for S:

`S_n=(5(2^(16)-1))/(2-1)\rightarrow S_n=327675`