We see that the common ratio is 2.5. We have the equation being:
[tex] a_k=19.2*(2.5)^k [/tex]
so:
[tex] 1875=19.2*2.5^k \implies \\
97.65625=2.5^k\implies\\
k=\log_{2.5}{97.65625} = 5 [/tex]
So, we can use the geometric series formula to get:[tex] \sum_{k=1}^{5}a_k=19.2(2.5+2.5^2+2.5^3+2.5^4+2.5^5) \\
2.5+2.5^2+2.5^3+2.5^4+2.5^5 = S\\
2.5^2+2.5^3+2.5^4+2.5^5+2.5^6=2.5S\\
2.5^6-2.5=1.5S\\
\frac{2.5^6-2.5}{1.5}=S\\
19.2*\frac{2.5^6-2.5}{1.5}=3093 [/tex]
