Question

an = a, .m-1 5. For the series (5 + 10 + 20 + ... + 20480): Calculate the 10th term in this series. =? a = ? Calculate the sum of the given series. Sn a,(1-r") 1=r

Answer 1

We can find the common ratio with the following formula:

`r=(a_(n+1))/(a_n)`

In this case, we have the following:

`\begin{gathered} r_1=(a_2)/(a_1)=(10)/(5)=2 \\ r_2=(a_3)/(a_2)=(20)/(10)=2 \end{gathered}`

we can see that the common ratio is r = 2. Then, we have the following formula for the sequence:

`\begin{gathered} a_n=a_1\cdot r^(n-1) \\ a_1=5 \\ r=2 \\ \Rightarrow a_n=5\cdot2^(n-1) \end{gathered}`

Now, to find the 10th term,we make n = 10:

`\begin{gathered} n=10 \\ \Rightarrow a_(10)=5\cdot2^(10-1)=5\cdot2^9=5\cdot512=2560 \end{gathered}`

therefore, the 10th term is 2560

The sum of the series can be calculated with the formula:

`\begin{gathered} S_n=(5(1-2^n))/(1-2)=-5(1-2^n) \\ \end{gathered}`