Question

The sum of an infinite geometric progression is 18 and the sum of the squares of the terms of the progression is 81. The first term and the common ratio of the geometric progression are, respectively:

• A. \( 7.2 \) and \( 0.6 \)
• B. \( 7.2 \) and \( 0.8 \)
• C. \( 3.6 \) and \( 0.6 \)
• D. \( 3.6 \) and \( 0.4 \)

Hint:

For infinite geometric progressions, use the sum formula \( \frac{a}{1 - r} \) and the sum of squares formula \( \frac{a^2}{1 - r^2} \) to solve for the first term and the common ratio.

Answer 1

The Correct Option is A

The sum of an infinite geometric progression is given by: \[ S = \frac{a}{1 - r} \] where \( a \) is the first term and \( r \) is the common ratio. We know that the sum \( S = 18 \), so: \[ \frac{a}{1 - r} = 18 \] Also, the sum of the squares of the terms is given by: \[ S_{\text{sq}} = \frac{a^2}{1 - r^2} = 81 \] By solving these two equations, we find that the first term \( a = 7.2 \) and the common ratio \( r = 0.6 \).