Writing Write a brief paragraph explaining why the terms of...

Writing Write a brief paragraph explaining why the terms of a geometric sequence decrease in magnitude when $-1<r<1$ Project: Housing Vacancies To work an extended application analyzing the numbers of vacant houses in the United States from 1990 through $2011,$ visit this text's website at LarsonPrecalculus.com. (Source: U.S. Census Bureau)

Writing Write a brief paragraph explaining why the
terms of a geometric sequence decrease in magnitude
when $-1<r<1$
Project: Housing Vacancies To work an extended
application analyzing the numbers of vacant houses in the
United States from 1990 through $2011,$ visit this text's
website at LarsonPrecalculus.com. (Source: U.S. Census
Bureau)

Key Concepts

Geometric Sequence

A geometric sequence is a series of numbers where each term is found by multiplying the previous term by a fixed, non-zero number called the common ratio. This type of sequence can either grow or decay, depending on the value of the common ratio.

Common Ratio

The common ratio is the constant multiplier between consecutive terms in a geometric sequence. When this ratio is between -1 and 1, it causes each term to be a fraction of the previous term in magnitude, leading to a decrease in the absolute size of the terms.

Convergence

A geometric sequence converges when the absolute value of the common ratio is less than one. This means that as more terms are generated, they become progressively closer to zero, which is why the sequence decreases in magnitude over time.

Magnitude

Magnitude refers to the absolute value of the terms in the sequence. Even if the ratio is negative, as long as its absolute value is less than one, the sizes of the terms will shrink, ensuring that the overall magnitude of the terms decreases as the sequence progresses.

Transcript

00:01 So this kind of problem is trying to get you to think about why a geometric sequence when the ratio is less than one or greater than negative.

00:09 When it's a little bit of a small value, why does that, how does that change the sequence? and i like to go into discussion about this because you have two different types of formulas for a geometric sequence.

00:21 You have for the sum of a geometric series.

00:24 I mean, for a finite sequence, you have this formula and for an infinite sequence, of this formula is much more simplified.

00:31 It really just drops the numerator from it.

00:33 And the reason why is because a infinite sequence will eventually max out.

00:40 Okay, and i'm going to kind of show you with some examples, but that could be.

00:43 So a finite, as a geometric series that's finite, is when the ratio, the absolute value of the ratio is greater than one.

00:51 So let's say we had a sequence such as 2, 4, 8, 16, 32.

00:58 See how those numbers are the number numbers are key, they keep getting larger, right? an infinite sequence is when the absent value of the ratio is less than one.

01:06 It's a small value.

01:08 So that same sequence starting at two, instead of it going times two to four, let's do times a half.

01:14 So down to one, a half, a fourth, and then an eighth, and a sixteenth.

01:21 Like they keep getting smaller, but that one keeps getting larger...

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