Question

Find the value of the following expression and round to the nearest integer:

Answer 1

44234

Step-by-step explanation:
`\sum ^(97)_(n=1)80(1.03)^(n-1)`

The values of n is from 1 to 97.

We need to sum the result of each of the term from 1 to 97.

When n = 1

`\begin{gathered} =80(1.03)^(1-1) \\ =80(1.03)^0\text{ = 80(1)} \\ =\text{ 80} \end{gathered}`
`\begin{gathered} \text{when n = 2} \\ =80(1.03)^(2-1)\text{ = }=80(1.03)^1 \\ =\text{ 80(1.03) = 82.4} \\ \text{when n= 3} \\ =80\mleft(1.03\mright)^{\mleft\{3-1\mright\}}\text{ = }=80^{}\mleft(1.03\mright?^2) \\ =84.872 \end{gathered}`

Finding the sum:

`\begin{gathered} \text{The sequence is a geoemtric sequence:} \\ \text{common ratio = next term/previous term = 1.03} \\ r\text{ >1} \\ \text{Sum of sequence in geometric sequence when r>1:} \\ S_n\text{ = }(a(r^n-1))/(r-1) \end{gathered}`
`\begin{gathered} n\text{ = 97} \\ r\text{ = }(84.872)/(84)=\frac{\text{ 84}}{80}\text{ = 1.03} \\ a\text{ = first term = 80} \\ S_(97)\text{ = }\frac{80(1.03^(97)-1)}{1.03-\text{ 1}} \\ \end{gathered}`
`\begin{gathered} S_(97)\text{ = }(80(16.5878))/(0.03) \\ S_(97)\text{ =}44234.1333 \end{gathered}`

To the nearest integer, the sum of 97 terms of the geometric sequence given is 44234.