Consider an infinite geometric series with first term ' a ' and common ratio ' r '. If the sum of infinite geometric series is 4 and the second term is 3 4 then

Step-by-step solution

The sum of an infinite geometric series is given by S = a 1-r = 4 , where |r| The second term of the series is ar = 3 4 . From the first equation, a = 4(1-r) . Substituting this into the second equation: 4(1-r)r = 3 4 4r - 4r^2 = 3 4 16r - 16r^2 = 3 16r^2 - 16r + 3 = 0 Solving the quadratic equation using the quadratic formula r = -b b^2 - 4ac 2a : r = 16 256 - 192 32 = 16 64 32 = 16 8 32 This gives two possible values for r : r₁ = 24 32 = 3 4 and r₂ = 8 32 = 1 4 For r = 3 4 , a = 4(1 - 3 4 ) = 4( 1 4 ) = 1 . For r