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

Can anyone help me solve this? Please and thank you!-example-1

1 Answer

5 votes

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

User ZeeShaN AbbAs
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