Problem

Solve each application. (Hint: Immediately after …

Problem 47 Easy Difficulty

Solve each application. (Hint: Immediately after reading the problem, determine whether you
need to find a specific term of a sequence or the sum of the terms of a sequence.)
A particular substance decays in such a way that it loses half its weight each day. In how many days will 256 g of the substance be reduced to 32 g? How much of the substance is left after 10 days?

Answer

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Precalculus

Beginning and Intermediate Algebra

Chapter 14

Sequences and Series

Section 3

Geometric Sequences

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Video Transcript

Okay. This question is asking is about a substance which decay so that lose half of its way each day. Okay, we're starting out with 256 grams of the substance. Okay, so after one day, we know that there will be 256 times 1/2 which is 128 grands play. After the second day, there will be 128 times 1/2 which is 64 brands. We continue this pattern on as we need to. We're being asked two questions here. The first questions it is in how many days? Well, to be 32 grams. Okay, well, if we just continue our pattern on one more, we actually land on. Exactly. We want 64 times 1/2 gives us 32 grams. But the answer to this is clearly three days. Just carrying on the pattern, we could see its forming a geometric sequence. How much of the substance is left after 10 days? Okay, we probably don't want toe map this out for 10 days, so we're gonna come up with a formula for this. We know the first time in this sequence is 256 and the common ratio that motor playing by is 1/2. So we know that any time in this sequence is 256 times 1/2 to the power in minus one. Okay, now, this is technically ah, we're actually said I wouldn't correct myself there. Right? You're gonna use 128 is the first time in this sequence. So the end value lines up with the days that we've been using. So this was one day today, three days. So we're gonna forget about 2 56 initial amount and treat. 128 is our day one amount. With this, we can find the amount on the 10th day. I just substituting in the number 10 everywhere we haven't end. So 128 times, 1/2 to the power of nine, which should give us 0.25 or 1/4 of the grand. Okay, so just be careful with how you have this set up working out which one you want to use is your first term said that this end matches the day that Europe

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