welcome to lesson 2 read 10 the geometric sequence a geometric sequence is a sequence in which we can get the succeeding term by multiplying the previous term to the fixed number called the common ratio in tagalog next term that we multiply not the new previous term there was a fixed number common ratio example time versus 2 6 and 18 where time fixed number i'm going to divide it ion which is having that in our common ratio and we're going to divide the second term so this is our a1 the first term a2 and a3 common ratio a2 divided by a1 so that means in arithmetic sequence moco had at the new common difference by subtraction terms even multiply like nothing new commodities a previous term so they thought you are not in the common ratio is six over two nine six divided by two that is three okay current number one x terms what about nothing 18 times three so 18 times 3 that is 8 16 24 carry 2 3 times 1 plus 2 so when x is 54. so now you'll see a four pharma quantity eight five would be fairly that let's see fifth important three so 54 times 3 that is 12 carry 1 15 plus 1 160 100 what if your teacher asks you to break the foreign paragraph formula to get the n term we have a n is equal to a 1 times r raised to n minus 1. a n is the inaudible the answer a1 that means the first term r is the common ratio and n is times r or nothing which is three three raised to n minus one that is nine minus one so 2 times 3 raised to 9 minus 1 that the is so multiply that in two times three raised to eight so first i come up with a calculator all to say by this is three times three that is nine times three twenty seven times three eighty one so eighty one times three pound fifth time nothing eighty one times three that is three eight six twenty four two hundred for three about fifth okay next in e5 a6 times 3 so 3 times 3 3 times 3 is [Music] 9 3 times four twelve here one three times two six plus one seven two nine six next eight seven tile times three armor a seven nine times three twenty seven carry two three times two six plus eight plus two eight next seven takes three twenty one so 24 25 36 carry two three plus two five and six six five six one six five six one championship three ways to eat next two times six five six one so multiply that ten six five six one times two so two six times two twelve and everyone so ten is five ten plus one eleven six times two twelve so our eight nine is twelve thousand one hundred twenty one calculator so that's a favorite io 163 times three parameters a6 163 times three so that is four eight six four eight six and we will see e6 next times four eight six times three one four five eight anions e7 next one four five eight times three four three seven four different volumes at last times three thirteen thousand one hundred twenty two yeah what is that i the gamma types about multiplied so five times two ten eleven everyone so six times two twelve plus one thirteen thirteen sorry for that sorry region of a2 the second term divided by a1 the first term so let's see geometric sequence and now let's proceed to geometric mean it is a term between any two terms in geometric sequence okay example trial find the geometric mean between two and eight so maritime formula john we have geometric mean is equal to the square root of the two given term of the sequence so it got two and eight so two times e all right so that is i square root of a times b that would be our formula multiply that in 2 times 8 that is 16 square root of 16 that is 4. so i'm going to let you geometry near i'll try any other examples from the geometric sequence 2 4 8 16 and 32 find the geometric mean between the following first and fifth terms so that's the first term can i do that's the fifth term first term second third third third fourth element fifth third thirteen so geometric mean is equal to the square root of two terms multiply 2 times 32 square root of 2 times 32 64. and the square root of 64. that is e so next fourth and then term so halabad is fourth first ratio theta so above ratio not in a 2 divided by a 1 so 4 divided by 2 that is 2. so according to the external beta 32 times 2 that is 64. 64 times 2 120 e 128 times 2 that is 256 256 thanks to 500 well i got one two three four five six seven eight nine so 512 times two that is one thousand twenty four and now enter nothing one thousand twenty four now here the geometric mean square root of sixteen times one thousand twenty four multiply nothing sixteen times one thousand twenty four a perfect square number gonna see one thousand thirty four so that is sixteen times square root of 1024 square root of 16 times the square root of 1024 square root of 16 is 4 times the square root of 1024 is 32. so final answer now then 4 times 32 that is 120 so ayan will see geometric mean and now let's proceed to the geometric series geometric series it is the sum of our geometric sequence let us have our first example what is the sum of the first eight terms of the geometric sequence 1 3 9 27 so we're going to do something here so it is happening you smell some of the geometric scene friends now the time formula we have sn is equal to a1 times r raised to n minus 1 over r minus 1 wherein a1 is the first term r is the common ratio and n is the nth term so sn we're not doing a term so that is c as a is equal to a1 and then conveniently the first term which is one r is the common ratio we know a2 divided by a1 3 divided by one that is three and minus one raised to n that is a then term minus one all over r nathan is three minus one so since one monitor when additionally three raised to eight three times three nine times three twenty seven times three eighty one eighty one times three two hundred so 81 times 3 that is 3 86 243 lima times 3 9 12 31 6 7 2 9 12 30 27 32 6 7 8 7 3 21 next seven and one eight nine two one eight seven times three so we have one carry two eight sixteen twenty four plus two twenty five twenty six carry two three five and six six five six one you'll see six five six one minus one over three minus one minus two next six five six one minus one that is six five six zero divided by two so divide that in cha six five six zero divided by two 6 divided by 2 3 so 6 bring down 5 5 divided by 2 so two four subtract one and six so three two eight three two eight so you can know your new adding sequence 81 81 times 3 243 243 times 3 7 9 7 2 9 3 2 1 8 7 1 1 3 4 5 6 7 8 243 81 27 9 3 and 1 so 7 plus 9 16 plus 9 plus 3 19 plus 120 27 27 36 36 37 30 39 40 40 so 34 so four plus eight twelve plus two fourteen plus four [Music] eighteen plus eight twenty six plus two twenty eight thirty two two plus one three plus seven ten plus two wow then carry one one plus two that is geometric sequence formula so eonc geometric series and we also have the infinite geometric series that means geometric sequences so we have the formula s infinity is equal to a 1 over 1 minus r example diode what is the sum of the geometric sequence eight negative two one-half negative one over eight and uh ellipsis sequence again so completely nothing happening formula so s infinity is equal to an a1 not n so given which is 8 over 1 minus r naught which is the common ratio so what happens so r naught indeed this equation is a negative two over a one which is a 2 divided by 8 can be written as can be written in c plus 1 so negative 1 4 so minus negative 1 4. so 8 over 1 negative times negative that will become positive 1 4. so eight over one plus one four so lcd is four four times one four plus copy one now we have eight over four plus one five over four fraction i o so eight times the reciprocal of the denominator four over five so eight times four eight sixteen twenty four thirty two thirty two over five so infinite geometric sequence a famous example geometric series is the pendulum familiar for example on the first swing the length of the arc through which a pendulum swings is 10 decimeter the length of each successive swing is two-thirds of the preceding term or preceding swing find the total distance it has traveled before coming at rest so did there be nothing to all be given at the end so maryland the first swing yeah i then decimeter so you're making a1 and the length of each successive swing is to turn [Music] r nothing is 2 3 2 over so 10 over 3 3 x 1 the minus so then over 3 minus 2 is 1 copy the numerator times the reciprocal of the denominator 3 over 1. so then takes me that is thirty now you need that in the m so travel channel of thirty two thirds infinite geometry series