Hello mga kawawmat! In previous lesson, you already know sequences in which a certain number is added to each term to get the next term. Now, you will explore sequences in which a certain number is multiplied.
So in this video, we define geometric sequence and also we identify the common ratio and the next term of the sequence, find the n-term of the geometric sequence, and solve problems involving geometric sequence. We're going to understand how to get the ratio, the concept of ratio for this type of sequence. So let's have a short activity. Activity.
Find the ratio of the second number to the first number. For example number 1, we have 2 and 8. So ang gagawin lang natin para makuha natin yung ratio ng dalawa, i-divide lang natin. Yung second number dun sa first number.
So that will be 8 divide 2. So therefore, yung ratio nga is 4. For number 2, we have negative 3 and 9. So divide lang natin yung 9 kay negative 3. So yung ratio nito ay negative 3. For number 3, we have 1 and 1 half. So i-divide lang natin si 1 half kay 1. So the ratio is 1 half. A sequence is geometric if there exists a number r called the common ratio. So we represent common ratio, the small letter r. The common ratio r can be determined by dividing any term in the sequence by the term that precedes it.
Okay, let's have an example. identify the common ratio and the next term in the following sequences. For number 1, we have 1, 2, 4, and 8. Of course, kanina sa first activity natin, diniscuss ko na kung paano kukunin yung common ratio nyan. So, ang gagawin lang, Yung second term, i-divide dun sa first term. So, 2 over 1. So, yung common ratio ng sequence neto is 2. So, therefore, pagkukuni natin yung next term, i-multiply lang natin si 8. I-multiply natin si 2 kay 8 para makuha yung next term.
So, the next term is 16 since 8 times 2 is equal to 16. For number 2, we have 80, 20, and 5. Same process. Para makuha natin yung common ratio, i-divide lang natin yung second term natin sa first term. So, 20 over 80. that is 1 fourth. So, para makuha yung next term, imumultiply natin yung 5 kay 1 fourth. So, that is, the next term is 5 over 4. Since 5 times 1 fourth, so that is 5 times 1 is equal to 5 all over 4. Okay, for number 3, we have 2, negative 8, 32, and negative 128. Same process to get the common ratio.
Didivide lang natin yung second term sa first term. So that is negative 8 over 2, that is equal to negative 4. So therefore, yung next term natin, imumultiply natin si negative 128 kay negative 4. And that is 512. Okay, you all know that the geometric sequence, so malilaman lang natin ang sequence na geometric kapag may common ratios nila. So I'll give you more examples to identify the given sequences if this sequence is geometric or not. Okay, I have four examples here.
So, for the first example, we have 5, 20, 80, 320. Of course, para ma-check nyo na talagang may common ratio sila, so ang gagawin natin, kunin natin yung mga common ratio nila. So, una muna natin i-check. Pag dinivide natin yung 20 kay 5, and the answer is 4. Yung 80, didivide kay 20, the answer also is 4. 320, i-divide kay 80, the answer is also 4. So, nakikita nyo na same lahat yung sagot.
So, therefore, meron silang common ratio. So, this is adjuvant. Metric sequence. Another.
7 square root of 2, 5 square root of 2, 3 square root of 2, and square root of 2. So to check, pag dinivide natin si 5 square root of 2 kay 7 square root of 2, ang makukuha natin dyan is 5 over 7 kasi makakancel natin to. Si 3 squared of 2, pag dinivide natin kay 5 squared of 2, ang makukuha natin is 3 over 5. So dito pa lang makikita nyo na magkaiba ang sagot, yung ratio nila. So hindi common. So therefore, yung number 2 is not a geometric sequence.
For number 3, we have 5, negative 10, 20, and negative 40. So same process, divide natin yung second term sa first term to check kung may common ratio sila. Negative 10 divided by 5, that is negative 2. 20 divided by negative 10, the answer is negative 2. At yung negative 40 divided by 20, the answer also is negative 2. So, therefore, meron silang common ratio. So yung number 3 natin is a geometric sequence For number 4, we have 10 over 3, 10 over 6, 10 over 9, and 10 over 15 So checkin muna natin kung may common ratio yung given sequence So kapag dinivide natin yung second term sa first term natin, ang sagot doon is 1 half Yung 10 over 9 over 10 over 6, ang sagot ay 2 third Ang 10 over 15 over 10 over 6 over 9, ang sagot ay 3 fifths.
So, magkaiba ang ratio nila. So, therefore, number 4 is not a geometric sequence. The n-term of a geometric sequence is given by a sub n.
So, tandaan yung formula na ito. Is equal to a sub 1 times r raised to n minus 1. Take note, r should not be equal. to zero.
So, yung a sub 1 natin dito, that is our first term. And yung r natin is yung common ratio natin. Yung n, that is the number of terms.
Nung kinukuha. Okay. Okay.
Let's try to answer this example. What is the 10th term of the geometric sequence given 8, 4, 2, and 1? So, identify muna natin yung mga given.
So, yung common ratio natin, so 4 over 8, that is 1 half. Yung first term natin, that is 8. So, we are using this formula. So, ang gagawin lang natin is sa substitute yung mga nakuha nating value sa common ratio.
Yung first term at yung... yung n natin is yung 10. Okay. So, a sub 1, that is 8 times 1 half raised to 10 minus 1. Bakit 10? So, we have 10 term, yung n natin. And then, 8 times 1 half, so 10 minus 1, is 9. Then, 1 half raised to 9. So, ang gagawin nyo muna, ito muna yung kailangan nyo yung...
evaluate yung 1 half raised to 9 hindi yung 8 times 1 half kailangan yung 1 half raised to 9 muna so that is 1 half raised to 9 so that is 1 times Raise to 9 is 1. 2 raised to 9 is 512. And then multiply. That is 8 over 512. And then 8 over 512 is, malo lowest term natin kasi divisible sila sa 8. So that is, the final answer is 1 over 64. Okay. So madali lang using the formula. So I have here another set of activities.
or exercises. Find the missing term in 3, 12, 48. So, ang gagawin muna natin, of course, kunin natin yung common ratio para madali natin ma-identify kung ano yung missing term. Since alam na natin yung common ratio, Ratio natin, so therefore, yung last na term dito, ito mo multiply lang natin kay 4 para makuha natin. And that is 48 times 4, that is 192. So ang kasunod dun ay 192. So ano kaya yung kasunod ni 192?
Same process, times kay 4, so the answer is 768. Another. Find the missing term in? Okay. Blank, blank, 32, 64, 128. So, ang hinahanap naman yung dalawang nasa unahan.
So, paano natin gagawin? Okay, madali lang yan. Ang gagawin nyo, kunin ulit yung common ratio natin. So, that is 64 over 32. That is 2. Since, pabaliktad yung gagawin natin, yung kukuni nating terms, bago kasi, no? Kasi pagpaganon, multiplication ang gagawin natin, yung mumultiply nyo.
So, pabaliktad yung gagawin natin. I-divide natin si 32 kay 2 para makuha ito. Ito muna yung kukuni nyo.
So, 32 divide 2, 16. So, 16 to. And then, 16 divide... 2, that is 8. So, yung first term dito is 8 and yung pangalawa, 16. So, kapag yung nasa unan yung hinahanap, ang gagawin nyo lang, i-divide nyo kapag nakuha nyo yung common ratio. Okay, I have here one problem, okay, wherein the geometric sequence na pwede natin i-apply. During the initial phase of an outbreak of measles, the number of infections is less than 1. can grow geometrically.
If there were 4, 8, 16 on the first 3 days of an outbreak of the measles, how many will be infected on the 6th day? Okay, gamit yung formula, unang munang gagawin, identify the given. So, yung common ratio natin, 8 over 4, that is 2. And then yung first term natin is 4. And then yung n natin is 6. So gamit yung formula, substitute the value of the first term is 4. And then this is the common ratio, raised to 6 minus 1. And then 4 times 2 raised to 5. And 2 raised to 5 is 32. 32 times 4, that is 128. Ibig sabihin, there will be 128 people infected with measles on the 6th day.
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