Good day welcome to our online math tutorial series of math lessons for grade 10 this is great and mathematics for quarter 1 module 5 geometric sequences versus arithmetic sequences this is teacher marian m guerrero your math teacher for today before we start please prepare the.
Following your math module piece of paper a pen and your notebook in this lesson you are expected to determine a geometric sequence identify the common ratio of a geometric sequence find the next remove a geometric.
Sequence and differentiate arithmetic sequence and a geometric sequence let us start our lesson with an activity am i arithmetic or not so you have to identify whether the given sequence is arithmetic or not write a s if it is an arithmetic sequence otherwise write an a remember.
Last week we have discussed all about arithmetic sequence and we have figured out on how to determine if a sequence is arithmetic or not number 1 5 11 17 23 to determine if it is an arithmetic sequence we have to get its common difference so in finding the common difference we.
Are to subtract a term to its receiving term for example common difference is eleven minus five is six another seventeen minus eleven is also equal to six the two common differences are equal therefore this is an arithmetic sequence.
For number 2 3 9 27 81 let us try to get a common difference subtract the second term to its first term 9 minus 3 is 6 and when we subtract the third term to its second term 27 minus 9 is 18. notice that the differences of the two are not equal.
Therefore this is not an arithmetic sequence number three three six nine twelve let us determine the common difference by subtracting the second term to its first term six minus three is three and also subtract the third term to its.
Second term nine minus six is also equal to three there is a common difference of three therefore this is an arithmetic sequence number four five ten twenty forty subtract the second term to its first term that is five.
And subtract the third term to its second term twenty minus ten is ten notice that the differences are not equal therefore this is not an arithmetic sequence number 5 125 25 5 notice that when we subtract the two the.
Two terms 25 minus 125 is negative 100 and 5 minus 25 is negative 20. they are not equal therefore this is not an arithmetic sequence notice that only number 1 and number 3 are arithmetic sequence what kind of sequence are the others for number 2 number 4 and number five what kind of sequences are this.
For us to know more about geometric sequence let us consider the problem below christmas sun is approaching anna is planning to buy a new blouse which cost 300 pesos as a gift to her mother she started saving money on the first week of november and doubled the amount.
To be saved every week if she started saving five pesos on the first week will she be able to buy the blouse at the end of the second week of december in this problem let us have this table for the first row the number of weeks starting from the week 1 of november up to the second week.
Of december and for the second row we have the weekly savings of anna for the first week anna saved five pesos and for the next week it will be double the amount thus anna's saving is five pesos times two that is ten pesos.
For the third week of november doubling the amount of the last week 10 pesos times 2 that is 20 pesos and for the 4th week 20 pesos times 2 is 40 pesos next for the first week of december doubling the amount from the last week 40 times 2 is 80 pesos and for the last week the second week of.
December that is 80 80 pesos times two that is 160 pesos given all the savings of anna from week one of november to the second week of december the total savings now is 315 pesos so the amount obtained of 315 pesos therefore anna will be able to buy the blouse at the end of second week of.
December notice that the sequence formed here in the weekly savings of anna 5 10 20 40 80 160 this is an example of a geometric sequence and what is a geometric sequence a geometric sequence it is also known as geometric progression.
It is a set of terms in which each term after the first is obtained by multiplying the preceding term by the same fixed number called the common ratio which is commonly represented by r given the sequence earlier the savings of anna 5 10 20 40 81 60 notice that each term in the sequence is.
Obtained by multiplying a constant term and that number is 2 which is doubling the amount of receive of her savings in each week 5 times 2 is 10 10 times 2 is 20 20 times 2 is 40 40 times 2 is 80 and 80 times 2 is 160. this fixed number 2 is what we call the common ratio.
And we can obtain the common ratio by dividing a term to its receiving term for example 10 divided by 5 is 2 20 divided by 10 is 2 40 divided by 20 is 2 that is the same as for the others 40 divided by 80 divided by 40 is 2 and also 160 divided by 80 is 2.
There is a common ratio of 2. therefore it is a geometric sequence let us now have some example find the common ratio and the next term of the sequence 3 12 48 for this we have to find or identify the common ratio we can identify the common ratio by dividing the second term to its first term.
And that is 12 divided by three which is equal to four from this multiply the common ratio to the preceding term to get the next term since we are given the common ratios 4 and the last term given is 48 let us multiply 48 times 4 will give us 192 therefore the next term in the sequence is 192.
Another example find the common ratio and the next three terms of the sequence 2 6 18 54. first identify the common ratio we can identify it by dividing the second term to its first term 6 divided by two is equal to three thus the common ratio is three.
We will use this to multiply to the preceding term to get the next term to get the fifth term that is 54 times 3 which is 162 sixth term 162 times 3 is 486 and for the seventh term 486 times 3 is 1458. therefore the next three terms of the sequence are 162.
486 and 1458. for our next example find the common ratio and the next two terms of the sequence 400 200 100 notice that in this sequence it is descending in order thus when we identify the common ratio we have to divide the second term to its first term 200 divided by 400 that is.
When we simplify we will come up with one half thus the common ratio is one half then multiply the common ratio to the preceding term to get the next term therefore a sub 4 is equal to 100 times one half that is 50 and the a sub 5 is equal to 50 times one-half that is 25.
Meaning the next two terms in the sequence are 50 and 25 this in our lessons from the last week we can now compare and differentiate an arithmetic sequence and a geometric sequence for arithmetic sequence we are adding a constant term.
For geometric sequence we are multiplying a constant term also arithmetic sequence has a common difference while a geometric sequence has a common ratio and in getting the next term of an arithmetic sequence we are subtracting a.
Term to its preceding term while for the geometric sequence we are dividing the second term to its first term let us now have some examples determine whether the sequence is arithmetic or geometric 12 15 18 21 we have to check if there is a common.
Difference or a common ratio exist for the common difference we will subtract the second term to the first term that is 15 minus 12 which is three also let us also subtract the third term to its second term eighteen minus fifteen is also equal to three notice that there is a common difference.
Of three now let us check if there is a common ratio to get the common ratio we will sub we will divide the second term to its first term and that is 15 divided by 12 when we simplify this that is 5 over 4 and also dividing the third term to the second term eighteen all over fifteen.
Which is six all over five since five fourths is not equal to six fifths then there is no common ratio it means the sequence is an arithmetic sequence for the last example determine whether the sequence is arithmetic or geometric now 5 15 45 check if a common difference or a common.
Ratio exists for the common difference subtract the second term to the first term 15 minus 5 is 10 and then subtract also does the return to the second term 45 minus 15 is 30. notice that the two differences are not equal therefore there is no common difference.
For the common ratio divide the second term to the first term 15 divided by 5 is 3 also divide the third term to the second term 45 divided by 15 is also equal to 3. since the the two ratio are equal then there is a common ratio of three meaning.
The sequence is a geometric sequence given the examples that we've had in our lesson let us now synthesize our discussion first a geometric sequence is a set of terms in which each term after the first is obtained by multiplying the preceding term but by the same fixed number called the common ratio.
A common ratio is obtained by dividing a term to its preceding term and the terms in arithmetic sequence obtained by adding a common difference to its preceding term while the terms in a geometric sequence is obtained by multiplying a common ratio to its preceding term with that.
That is all for today thank you for listening