The sum of an infinite number of terms of GP is 4 and the sum of their cubes is 192. Find the series.
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lets say
a= the first term
r =common ratio
so formula for the sum of infinite geometric progression: a/1-r= 4
The sequence which consist of cubes is geometric progression in itself.
so the first term will be= a³ and common ratio r³
the equation will be
a³/1-r³=192
dividing both the equations
a²/ 1+r+r²= 48
now this equation can be expressed as: a= 4 (1-r)
substituting in the above equation
4² (1-r)²/ 1+r +r² = 48
(1-r)²= 3(1+r+r²)
1-2r +r² = 3r² + 3r + 3
a= the first term
r =common ratio
so formula for the sum of infinite geometric progression: a/1-r= 4
The sequence which consist of cubes is geometric progression in itself.
so the first term will be= a³ and common ratio r³
the equation will be
a³/1-r³=192
dividing both the equations
a²/ 1+r+r²= 48
now this equation can be expressed as: a= 4 (1-r)
substituting in the above equation
4² (1-r)²/ 1+r +r² = 48
(1-r)²= 3(1+r+r²)
1-2r +r² = 3r² + 3r + 3
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