[tex]\bf \qquad \qquad \textit{sum of a finite geometric sequence} \\\\ \displaystyle S_n=\sum\limits_{i=1}^{n}\ a_1\cdot r^{i-1}\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=\textit{last term's}\\ \qquad position\\ a_1=\textit{first term}\\ r=\textit{common ratio}\\ \cline{1-1} n=9\\ r=3.8\\ \stackrel{S_n}{S_9}=44240 \end{cases}[/tex]
[tex]\bf 44240=a_1\left( \cfrac{1-3.8^9}{1-3.8} \right)\implies 44240\approx a_1\left( \cfrac{-165215.101}{-2.8} \right) \\\\\\ 44240\approx a_1(59005.393)\implies \cfrac{44240}{59005.393}\approx a_1\implies \stackrel{\textit{rounded up}}{0.75=a_1}[/tex]
When all the terms of a geometric sequence are added, then that expression is called geometric series. The first term of the given geometric series is 0.75.
When all the terms of a geometric sequence are added, then that expression is called geometric series.
Let's suppose its initial term is, the multiplication factor is r
and let it has total n terms, then, its sum is given as:
[tex]S_n = \dfrac{a(r^n-1)}{r-1}[/tex]
(sum till nth term)
Given the sum of the geometric series is 44,240, while the number of terms is 9. Also, the common ratio of the series is 3.8. Thus, we can write,
Sₙ = a(rⁿ-1)/(r-1)
44240 = a(3.8⁹ - 1)/(3.8 - 1)
44240 = a 59005.3933
a = 0.75
Hence, the first term of the given geometric series is 0.75.
Learn more about the Sum of terms of geometric sequence:
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