Answer:
[tex]S_{1893} =5632.98[/tex]
Step-by-step explanation:
The correct form of the question is:
[tex]S = 1.001^{1892} + ... + 1.001^2 + 1.001 + 1[/tex]
Required
Solve for Sum of the sequence
The above sequence represents sum of Geometric Sequence and will be solved using:
[tex]S_n = \frac{a(1 - r^n)}{1 - r}[/tex]
But first, we need to get the number of terms in the sequence using:
[tex]T_n = ar^{n-1}[/tex]
Where
[tex]a = First\ Term[/tex]
[tex]a = 1.001^{1892}[/tex]
[tex]r = common\ ratio[/tex]
[tex]r = \frac{1}{1.001}[/tex]
[tex]T_n = Last\ Term[/tex]
[tex]T_n = 1[/tex]
So, we have:
[tex]T_n = ar^{n-1}[/tex]
[tex]1 = 1.001^{1892} * (\frac{1}{1.001})^{n-1}[/tex]
Apply law of indices:
[tex]1 = 1.001^{1892} * (1.001^{-1})^{n-1}[/tex]
[tex]1 = 1.001^{1892} * (1.001)^{-n+1}[/tex]
Apply law of indices:
[tex]1 = 1.001^{1892-n+1}[/tex]
[tex]1 = 1.001^{1892+1-n}[/tex]
[tex]1 = 1.001^{1893-n}[/tex]
Represent 1 as [tex]1.001^0[/tex]
[tex]1.001^0 = 1.001^{1893-n}[/tex]
They have the same base:
So, we have
[tex]0 = 1893-n[/tex]
Solve for n
[tex]n = 1893[/tex]
So, there are 1893 terms in the sequence given.
Solving further:
[tex]S_n = \frac{a(1 - r^n)}{1 - r}[/tex]
Where
[tex]a = 1.001^{1892}[/tex]
[tex]r = \frac{1}{1.001}[/tex]
[tex]n = 1893[/tex]
So, we have:
[tex]S_{1893} =\frac{1.001^{1892} *(1 -\frac{1}{1.001}^{1893})}{1 -\frac{1}{1.001} }[/tex]
[tex]S_{1893} =\frac{1.001^{1892} *(1 -\frac{1}{1.001}^{1893})}{\frac{1.001 -1}{1.001} }[/tex]
[tex]S_{1893} =\frac{1.001^{1892} *(1 -\frac{1}{1.001}^{1893})}{\frac{0.001}{1.001} }[/tex]
[tex]S_{1893} =\frac{1.001^{1892} *(1 -\frac{1}{1.001^{1893}})}{\frac{0.001}{1.001} }[/tex]
Simplify the numerator
[tex]S_{1893} =\frac{1.001^{1892} -\frac{1.001^{1892}}{1.001^{1893}}}{\frac{0.001}{1.001} }[/tex]
[tex]S_{1893} =\frac{1.001^{1892} -1.001^{1892-1893}}{\frac{0.001}{1.001} }[/tex]
[tex]S_{1893} =\frac{1.001^{1892} -1.001^{-1}}{\frac{0.001}{1.001} }[/tex]
[tex]S_{1893} =(1.001^{1892} -1.001^{-1})/({\frac{0.001}{1.001} })[/tex]
[tex]S_{1893} =(1.001^{1892} -1.001^{-1})*{\frac{1.001}{0.001}}[/tex]
[tex]S_{1893} =\frac{(1.001^{1892} -1.001^{-1}) * 1.001}{0.001}[/tex]
Open Bracket
[tex]S_{1893} =\frac{1.001^{1892}* 1.001 -1.001^{-1}* 1.001 }{0.001}[/tex]
[tex]S_{1893} =\frac{1.001^{1892+1} -1.001^{-1+1}}{0.001}[/tex]
[tex]S_{1893} =\frac{1.001^{1893} -1.001^{0}}{0.001}[/tex]
[tex]S_{1893} =\frac{1.001^{1893} -1}{0.001}[/tex]
[tex]S_{1893} =5632.97970294[/tex]
Hence, the sum of the sequence is:
[tex]S_{1893} =5632.98[/tex] ----- approximated
