Sum of geometric sequence?help?

Question

The sum of the first and second terms of a geometric sequence is 108 and the sum of the third and fourth terms is 12. Find the two possible values for the first term of the sequence.

Answer 1

The sum of the first and second terms of a geometric sequence :

a + ar  = 108

a (1 + r) = 108.

a = 108/(1 + r ) → ( 1 )

The sum of the third and fourth terms of a geometric sequence :

ar ² + ar ³ = 12

ar ² (1 + r ) = 12

r ²  * [a (1 + r )] = 12

Substitute a (1 + r ) = 108 in above equation.

r ²  * 108 = 12

r ² = 12/108

r ² = 1/9

r ² = (± 1/3)²

⇒ r = ± 1/3.

Substitute the values r = ± 1/3 in eq ( 1 ).

When, r = - 1/3, a = 108/(1 - (1/3)) = 108/(2/3) = (108 * 3)/2 = 54 * 3 = 162.

When, r = 1/3, a = 108/(1 + (1/3)) = 108/(4/3) = (108 * 3)/4 = 27 * 3 = 81.

The two possible values for the first term of the geometric sequence are 162 and 81.