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## Algebra 1

### Unit 9: Lesson 3

Introduction to geometric sequences- Intro to geometric sequences
- Extending geometric sequences
- Extend geometric sequences
- Extend geometric sequences: negatives & fractions
- Using explicit formulas of geometric sequences
- Using recursive formulas of geometric sequences
- Use geometric sequence formulas

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# Extending geometric sequences

CCSS.Math:

Sal finds the next term in the geometric sequence -1/32, 1/8, -1/2, 2,...

## Want to join the conversation?

- At0:13, why did he laugh? Did he say it wrong?(6 votes)
- because he said "sequenche" really funny apparently(17 votes)

- is there another way to explain this concept? So far, I've understood mostly everything except this?(4 votes)
- There maybe another way, but in order to help you, I need try to understand what you are having trouble with.

Have you watched the previous video, "Intro to geometric sequences"? Make sure you understand what geometric sequence is.

What is it that is confusing you in this video? Pause the video at the point where start to get lost, what time does it say on video progress line?(13 votes)

- is there a formula to figure out the common ration of a geometric sequence with fraction?(2 votes)
- To find the common ratio, you divide the next number by the current number. So lets say the common ratio is 1/2. a sequence starting from 32, 16, 8, 4, 2, 1, .5 ... You can divide starting anywhere, so 16/32 = 1/2, 8/16 = 1/2, 4/8 = 1/2, ...(13 votes)

- What is the General rule used to find the nth term from Geometric Sequence?(2 votes)
- a(n) = a * r^(n-1) ... I hope the formating is not confusing(4 votes)

- What is the recursive or explicit equation?(1 vote)
- how do I know what to multiply by? like where did he get the -4 from?(1 vote)
- The common ratio is found by dividing consecutive terms (second divided by 1st), so (1/8)/(-1/32) = (1/8)*(-32)=-4. Or (-1/2)/(1/8)= -1/2*6=-4, this pattern continues 2/(-1/2) = 2*-3=-4. Each give a common ratio of -4. so next is (2)(-4)=-8, then -8*-4=32, etc.(4 votes)

- How is -1/2 x -4 = 2?(0 votes)
- -1/2 x -4

= (-1)(1/2)(-1)(4) // expand negative number

= (-1)(-1)(1/2)(4) // commutative property of multiplication

= (1)(1/2)(4) // negative times negative equals positive

= (1/2)(4) // property of multiplicative identity element 1, eg (1)(a) = a

= (1/2)(4/1) // convert integer to rational number

= 4/2 // definition of multiplication of rationals

= 2 // 4 divided by 2 equals 2

.

.(5 votes)

- Couldn't you just divide each of the numbers in the sequench(haha) to the one before it to find the common ratio? is that an approved way to do it? I am always looking for more efficient ways to do things. It seems like Sal uses guess test and revise more than doing an additional equation to get it done.(1 vote)
- That is a good way to do it, always have to careful with a lot of fractions.(3 votes)

- I understand this conceptually, and it's easier with whole numbers for me. My question is, how can people so easily go from one term to another and just go "oh they're multiplying by 4/5ths" Is there a way to become more comfortable doing these types of problems when the fractions are involved?

In a later problem I think one of the terms goes from 26/16 to -9/4. Again, I understand how everything is working, but am struggling with problems like that to determine what the sequence is changing by each term. Thanks!(2 votes)- Since you are multiplying by a constant (even if it is a fraction) to get from one to the next, so lets do it algebraically. Let x be our common ratio. So 26/16 x = -9/4. We have two choices, one is to divide (which is normally how we find the ratio, second term divided by first term). Thus we have (-9/4)/(26/16). But going back to the equation created above, the opposite of a fraction is to multiply by the reciprocal, so x = (-9/4)(16/26) = (-9/4)(8/13) = -18/13. This also shows how to divide fractions, reciprocate the denominator and multiply.

Since 26/16 reduces to 13/8, it probably is something else, 27/16 makes more sense. Thus, you get (-9/4)(16/27) and by changing places, the equivalent is (-16/4)(9/27) = -4/3. This seems more probable.(1 vote)

- how do I use the standard calculator times by 2/3(1 vote)
- type in *(2/3) and it should be right. Also, it's okay if you want to ignore the brackets in this case because times by 2/3 is the same thing as times 2 and divide by 3(2 votes)

## Video transcript

- [Voiceover] So we're told
that the first four terms of a geometric sequence are given. So they give us the first four terms. And they say what is the
fifth term in the sequence. And like always pause the video and see if you can come
up with the fifth term. Well what we have to remind ourselves is for a geometric sequence
each successive term is the previous term
multiplied by some number. And that number we call the common ratio. So let's think about it. To go from negative 1/32,
that's the first term to 1/8, what do we have to multiply by? What do we have to multiply by? Let's see. We're going to multiply, it's going to be multiplied by a negative since we went from a
negative to a positive. So we're going to multiply by negative and there's going to be a one over let's see to go from a 32 to an eight, actually it's not going to be a one over. It's going to be, this is
four times as large as that. It's going to be negative four. Negative 1/32 times negative
four is positive 1/8. Just to make that clear. Negative 1/32 times negative four that's the same thing
as times negative four over one. It's going to be positive. Negative times a negative is a positive. Positive four over 32. Which is equal to 1/8. Now let's see if that holds up. So to go from 1/8 to negative 1/2 we once again would
multiply by negative four. Negative four times 1/8 is negative 4/8, which is negative 1/2 and so then we multiply
by negative four again. So, let me make it clear. We're multiplying by
negative four each time. You multiply by negative four again, you get to positive two. Because negative four over negative two, you can do it that way, is positive two. And so to get the fifth
term in the sequence, we would multiply by negative four again. And so two times negative
four is negative eight. Negative four is the common ratio for this geometric sequence. But just to answer the question, What is the fifth term? It is going to be negative eight.