Mathematics

Mathematics

Problem

A geometric progression has first term 9 and common ratio −3-3. Each term is multiplied by −13-\dfrac{1}{3} to obtain a new geometric progression. a. (i) Find the first term of the new progression. b. (ii) Find its common ratio.
Intuitive explanation

The key insight is recognizing that multiplying every term of a geometric progression by a constant creates a new geometric progression with the same common ratio, but a different first term. Here, you're scaling the entire sequence by −13-\frac{1}{3}, so find the new first term by applying that multiplier to 9, then observe that the common ratio stays unchanged at −3-3.

Intuitive explanation

The key insight is recognizing that multiplying every term of a geometric progression by a constant creates a new geometric progression with the same common ratio, but a different first term. Here, you're scaling the entire sequence by −13-\frac{1}{3}, so find the new first term by applying that multiplier to 9, then observe that the common ratio stays unchanged at −3-3.

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