Question

find the first term and common ratio​

Answer 1

Answer :-

a = 1 or 3 , r = 3/4 or 1/4

Solution :-

Let the first term of infinite GP be 'a' and 'r'

Second term = a₂ = ar = 3/4

==> ar = 3/4

==> a = 3/4r

Sum of the infinite GP = S = 4

Using Sum of terms infinite GP formula

==> S = a/(1 - r)

==> 4 = ( 3/4r ) / ( 1 - r)

==> 4( 1 - r ) = 3/4r

==> 4 - 4r = 3/4r

==> 4r( 4 - 4r) = 3

==> 16r - 16r² = 3

==> 0 = 16r² - 16r + 3

==> 16r² - 16r + 3 = 0

==> 16r² - 12r - 4r + 3 = 0

==> 4r(4r - 3) - 1(4r - 3) = 0

==> (4r - 3)(4r - 1) = 0

==> 4r - 3 = 0 or 4r = 1

==> 4r = 3 or or r = 1/4

==> r = 3/4 or r = 1/4

When r = 3/4

==> a = 3/4r = 3/4 * ( 3/4) = 3/4 * 4/3 = 1

When r = 1/4

==> a = 3/4r = 3/4 * (1/4) = 3/4 * 4 = 3

Therefore the first term of of GP is 1 or 3 and the common ratio is 3/4 or 1/4.