Question

What is the first term in a geometric sequence if the common ratio is −3 and the sum of the first five terms is 427?

Answer 1

so, we know that r = -3, and that the sum of the first 5 terms is 427 hmmmm

$\bf \qquad \qquad \textit{sum of a finite geometric sequence}\\\\ S_n=\sum\limits_{i=1}^{n}\ a_1\cdot r^{i-1}\implies S_n=a_1\left( \cfrac{1-r^n}{1-r} \right)\quad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ n=5\\ r=-3\\ S_5=427 \end{cases} \\\\\\ S_5=a_1\left( \cfrac{1-r^5}{1-r} \right)\implies 427=a_1\left( \cfrac{1-(-3)^5}{1-(-3)} \right)$

$\bf 427=a_1\left(\cfrac{1+243}{1+3} \right)\implies 427=a_1(61)\implies \cfrac{427}{61}=a_1\implies 7=a_1 \\\\\\ n^{th}\textit{ term of a geometric sequence}\\\\ a_n=a_1\cdot r^{n-1}\qquad \begin{cases} n=n^{th}\ term\\ a_1=\textit{first term's value}\\ r=\textit{common ratio}\\ ----------\\ r=-3\\ n=5\\ a_1=7 \end{cases} \\\\\\ a_5=7 (-3)^{5-1}\implies a_5=7(-3)^4$

and surely you know how much that is.