Question

Can anyone help me solve this? Please and thank you!

Answer 1

It's the sum of a geometric sequence.

Let's rewrite it a bit:

`\displaystyle\\\sum_(n=1)^(10)8\left((1)/(4)\right)^(n-1)=8\sum_(n=1)^(10)\left((1)/(4)\right)^n\cdot \left((1)/(4)\right)^(-1)=8\sum_(n=1)^(10)\left((1)/(4)\right)^n\cdot 4=32\sum_(n=1)^(10)\left((1)/(4)\right)^n`

And now let's calculate this sum
`\displaystyle \sum_(n=1)^(10)\left((1)/(4)\right)^n`:

`S_n=(a(1-r^n))/(1-r)\\\\a=(1)/(4)\\n=10\\r=(1)/(4)\\\\S_(10)=((1)/(4)\cdot\left(1-\left((1)/(4)\right)^(10)\right))/(1-(1)/(4))=((1)/(4)\cdot\left(1-(1)/(1048576)\right))/((3)/(4))=((1048575)/(1048576))/(3)=(1048575)/(3145728)=\\=(349525)/(1048576)`

Now let's calculate the initial sum:

`\displaystyle 32\cdot \sum_(n=1)^(10)\left((1)/(4)\right)^n=32\cdot (349525)/(1048576)=(349525)/(32768)\approx10.67`