Question

Geometric Sequence S = 1.0011892 + ... + 1.0012 + 1.001 + 1

Answer 1

`S_(1893) =5632.98`

Explanation:

The correct form of the question is:

`S = 1.001^(1892) + ... + 1.001^2 + 1.001 + 1`

Required

Solve for Sum of the sequence

The above sequence represents sum of Geometric Sequence and will be solved using:

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

But first, we need to get the number of terms in the sequence using:

`T_n = ar^(n-1)`

Where

`a = First\ Term`

`a = 1.001^(1892)`

`r = common\ ratio`

`r = (1)/(1.001)`

`T_n = Last\ Term`

`T_n = 1`

So, we have:

`T_n = ar^(n-1)`

`1 = 1.001^(1892) * ((1)/(1.001))^(n-1)`

Apply law of indices:

`1 = 1.001^(1892) * (1.001^(-1))^(n-1)`

`1 = 1.001^(1892) * (1.001)^(-n+1)`

Apply law of indices:

`1 = 1.001^(1892-n+1)`

`1 = 1.001^(1892+1-n)`

`1 = 1.001^(1893-n)`

Represent 1 as
`1.001^0`

`1.001^0 = 1.001^(1893-n)`

They have the same base:

So, we have

`0 = 1893-n`

Solve for n

`n = 1893`

So, there are 1893 terms in the sequence given.

Solving further:

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

Where

`a = 1.001^(1892)`

`r = (1)/(1.001)`

`n = 1893`

So, we have:

`S_(1893) =(1.001^(1892) *(1 -(1)/(1.001)^(1893)))/(1 -(1)/(1.001) )`

`S_(1893) =(1.001^(1892) *(1 -(1)/(1.001)^(1893)))/((1.001 -1)/(1.001) )`

`S_(1893) =(1.001^(1892) *(1 -(1)/(1.001)^(1893)))/((0.001)/(1.001) )`

`S_(1893) =(1.001^(1892) *(1 -(1)/(1.001^(1893))))/((0.001)/(1.001) )`

Simplify the numerator

`S_(1893) =(1.001^(1892) -(1.001^(1892))/(1.001^(1893)))/((0.001)/(1.001) )`

`S_(1893) =(1.001^(1892) -1.001^(1892-1893))/((0.001)/(1.001) )`

`S_(1893) =(1.001^(1892) -1.001^(-1))/((0.001)/(1.001) )`

`S_(1893) =(1.001^(1892) -1.001^(-1))/({(0.001)/(1.001) })`

`S_(1893) =(1.001^(1892) -1.001^(-1))*{(1.001)/(0.001)}`

`S_(1893) =((1.001^(1892) -1.001^(-1)) * 1.001)/(0.001)`

Open Bracket

`S_(1893) =(1.001^(1892)* 1.001 -1.001^(-1)* 1.001 )/(0.001)`

`S_(1893) =(1.001^(1892+1) -1.001^(-1+1))/(0.001)`

`S_(1893) =(1.001^(1893) -1.001^(0))/(0.001)`

`S_(1893) =(1.001^(1893) -1)/(0.001)`

`S_(1893) =5632.97970294`

Hence, the sum of the sequence is:

`S_(1893) =5632.98` ----- approximated