Answer:
19,531,248
Step-by-step explanation:
General form of a geometric sequence: [tex]a_n=ar^{n-1}[/tex]
(where [tex]a[/tex] is the first term and [tex]r[/tex] is the common difference)
Given geometric series: 8 + 40 + 200 + ... + 15625000
[tex]\implies a_1=8[/tex]
[tex]\implies a_2=40[/tex]
[tex]\implies a_3=200[/tex]
To find the common ratio [tex]r[/tex], divide consecutive terms:
[tex]\implies r=\dfrac{a_2}{a_1}=\dfrac{40}{8}=5[/tex]
Therefore:
[tex]\implies a_n=8(5)^{n-1}[/tex]
To find n when [tex]a_n=15625000[/tex] :
[tex]\implies 8(5)^{n-1}=15625000[/tex]
[tex]\implies (5)^{n-1}=1953125[/tex]
[tex]\implies \ln (5)^{n-1}=\ln1953125[/tex]
[tex]\implies (n-1)\ln 5=\ln1953125[/tex]
[tex]\implies n=\dfrac{\ln1953125}{\ln5}+1[/tex]
[tex]\implies n=10[/tex]
Sum of the first n terms of a geometric series:
[tex]S_n=\dfrac{a(1-r^n)}{1-r}[/tex]
Therefore, sum of the first 10 terms:
[tex]\implies S_{10}=\dfrac{8(1-5^{10})}{1-5}[/tex]
[tex]\implies S_{10}=19,531,248[/tex]
