The 2nd and 3rd terms of a geometric progression (GP) are -6 and 3 respectively. To find the sum to infinity of this GP, we need to know the first term and the common ratio. We can derive these from the given terms and then apply the formula for the sum to infinity of a GP.
Step 1: Find the common ratio (r)
The common ratio of a GP is found by dividing any term by the previous term. We'll divide the 3rd term by the 2nd term:
$ r = \frac{\text{3rd term}}{\text{2nd term}} = \frac{3}{-6} = -0.5 $
So the common ratio of this GP is -0.5.
Step 2: Find the first term (a)
To find the first term, we use the property that any term in a GP can be expressed using the first term and the common ratio. The 2nd term (a2) can be written in terms of the first term (a) as follows:
$ a2 = a \times r $
We already know a2 (-6) and r (-0.5), so we can solve for a:
$ -6 = a \times (-0.5) $
Rearranging and solving for a gives:
$ a = \frac{-6}{-0.5} = 12 $
So the first term of the GP is 12.
Step 3: Calculate the sum to infinity (S∞)
The sum to infinity of a GP can be found using the formula:
$ S∞ = \frac{a}{1 - r} $
This formula is valid only when the absolute value of the common ratio, |r|, is less than 1. In our case:
$ r = -0.5 $
The absolute value of -0.5 is 0.5, which is indeed less than 1. So we can apply the formula:
$ S∞ = \frac{12}{1 - (-0.5)} = \frac{12}{1 + 0.5} = \frac{12}{1.5} = 8 $
Therefore, the sum to infinity for the given geometric progression is 8.
