Question

The common ratio of a geometric series is 9, while the sum of the first 10 terms in the series is 435,848,050. What is the first term in the series?

Answer 1

\[a _{1} + a _{2} + a _{3} + a _{4} + a ^{5} + a _{6} + a _{7} + a _{8} + a _{9} +a_{10} = 435,848,0050\]
but you know the common ratio is 9
\[a _{1} + 9a _{1} + 9a _{2} + 9a _{3} + 9a _{4} + 9a _{5} + a _{6} + 9a _{7} + 9a _{8} +9a_{9} = 435,848,0050\] you can get doing this till...
\[a _{1} + 9a _{1} + 9^{2}a _{1} + 9^{3}a _{1} + 9^{4}a _{1} + 9^{5}a _{1} + 9^{6}a _{1} + 9^{7}a _{1} + 9^{8}a _{1} +9^{9}a_{1} = 435,848,0050\]
\[a _{1} * (1 + 9+ 9^{2}+ 9^{3}+ 9^{4} + 9^{5}+ 9^{6} + 9^{7}+ 9^{8} +9^{9}) = 435,848,0050\]

Answer 2

\[a_{1} = \frac{ 435,848,0050 }{ (1 + 9+ 9^{2}+ 9^{3}+ 9^{4} + 9^{5}+ 9^{6} + 9^{7}+ 9^{8} +9^{9}) }\]