1. The problem statement, all variables and given/known data
A geometric series has first term and common ratio both equal to ##a##, where ##a>1##
Given that the sum of the first 12 terms is 28 times the sum of the first 6 terms, find the exact value of a.
Hence, evaluate
[itex]log_{3}(\frac{3}{2} a^{2}+ a^{4}+...+ a^{58})[/itex]
Giving your answer in the form of ##A-log_{3} B##, where ##A## and ##B## are positive integers to be determined.
2. Relevant equations
Geometric series, logarithms
3. The attempt at a solution
For the first part, I have to write them as
##S_{12} =28 S_{6}##. Then I have to apply the formula for geometric sum...
Then I arrived at ##a^{13} -28a^{7} +27a=0##
Factoring give me ##a(a^{12}-28a^{6}+27)=0##
How should I solve for a in this case?
A geometric series has first term and common ratio both equal to ##a##, where ##a>1##
Given that the sum of the first 12 terms is 28 times the sum of the first 6 terms, find the exact value of a.
Hence, evaluate
[itex]log_{3}(\frac{3}{2} a^{2}+ a^{4}+...+ a^{58})[/itex]
Giving your answer in the form of ##A-log_{3} B##, where ##A## and ##B## are positive integers to be determined.
2. Relevant equations
Geometric series, logarithms
3. The attempt at a solution
For the first part, I have to write them as
##S_{12} =28 S_{6}##. Then I have to apply the formula for geometric sum...
Then I arrived at ##a^{13} -28a^{7} +27a=0##
Factoring give me ##a(a^{12}-28a^{6}+27)=0##
How should I solve for a in this case?
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