Question

Given a geometric series, multiply every term by the same nonzero constant c. The resulting series must also be geometric. True False b) A new series is obtained from a geometric series by taking the reciprocal of each term. The new series must also be geometric. True False

Answer 1

1. True
2. True

Explanation:

1. In the original series, the terms are ...

a1 + a1·r + a1·r² + ...

Multiplying each term by c, we get ...

(a1·c) + (a1·c)·r + (a1·c)·r² + ...

This new series is a geometric series with common ratio r and first term a1·c.

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2. In the original series, the terms are ...

a1 + a1·r + a1·r² + ...

Taking the reciprocal of each term, the new series is ...

(1/a1) + (1/(a1·r)) + (1/(a1·r²)) + ...

= (1/a1) + (1/a1)(1/r) + (1/a1)(1/r)² + ...

The new series is a geometric series with common ratio 1/r and first term 1/a1.