hi guys hi guys hi guys hi guys hi guys it's me teacher in today's video hi guys it's me teacher going in today's video we will talk about the geometric means last time maritime video about the geometric sequence the answer of geometric sequence and right now focus data is to talk about the geometric means so by the way guys what does it mean a capacitor geometric means geometric means these are the terms between the first term and the last term of a given geometric sequence so without further ado let's let's stop it so we have here the formula so i know you're familiar with this formula i think dynamic netting's n term of geometric sequence which is a sub n is equal to a sub one times r raised to n minus one into quality formulae your a sub n is the nth or last term your a sub 1 is the first term your r is the common ratio your n is the number of terms now for this problem or for this topic i have three different examples for us to solve so let's try this one find the geometric means or geometric mean of the following so in number one we are given five blank twenty so we need to find a single geometric mean and this one is two black black and 686 we need to input or insert two geometric terms or two terms in this sequence and number three we are given three blank blank and one over nine so let's start with number one for number one if you're trying to solve for a single geometric mean there are two possible ways and how to do it again there are two possible ways on how to do it so i'm not a second term demand so that is a sub 2. what will i do is i will multiply the first and the third or i will multiply these numbers you have five times 20 and i will get the square root of their product okay so a sub 2 is equal to 5 times 20 that is the square root of 100 and getting the square root of 100 that is 10 so a sub 2 is equal to 10. i think second term method or in geometric meaning between 5 and 20. so this is 10. now this is the first method again if you're trying to solve for a geometric mean a single geometric mean and all you need to do is to multiply the first and last and then get the square root that's it okay zero python is saying in your second process method for the second process we will be using this formula a sub n is equal to a sub one [Music] times r raised to n minus one the last term guys is twenty so you have your prime t okay your a sub y is the first time which is five times your r which is what happened so that is r raised to n so we have one two and three very times three terms so three minus one and then we have here twenty is equal to five times r raised to two because we have three minus one here again three minus one title segment is three r squared so we will manipulate the equation by dividing both sides by five and this one is also by five cancel cancel we have here r squared and twenty divided by five is four and antarctica i can do guys is to find the common ratio and we need to get the square root of r squared and square root of four your r is equal to two it is a minute in combination again using this common ratio t x for example we will erase this we know that the common ratio is true okay the hormone ratio is two the first term is five but given your definition for the geometric sequence to find the next term you need to multiply the existing term by the common ratio which is two so five times two that will give you the answer of ten okay though we use the different methods same answer enough why not then okay first method that then if you're trying to uh get the geometric mean or a single geometric mean okay let me clarify that let's continue with item number two [Music] final number two guys we're given two black black and 686 now guys for this topic for this example we are asked to find two missing terms so we will use this formula a sub n is equal to e sub 1 times r raised to n minus 1 again your a sub n is the last term this term your a sub n is equal to 686 your a sub one is two because that is your la first term your n is the number of terms one two three and four your n is equal to four so plug in nothing values theta this is 686 is equal to your a sub 1 which is 2 your r is unknown your n is 4 that is 4 minus 1. simplifying this exponent you have 686 it's equal to 2 raised to r are 2 times r raised to 3 because 3 4 minus 1 is 3 and manipulating the equation divide both sides by 2 cancel cancel your head here r2 okay and then dividing 686 by 2 that is 343 now to get the common ratio get the cube root of r cube and the cube root of 343 r cube cube root of r r cube is r this is your common ratio and a cube root of 343 is seven okay so not in the second term a sub two you need to multiply the first term two by your r which is 7 and your a sub 2 is equal to 14 this is 14. and for a sub 3 get the second term which is 14 times 7 what is 14 times 7 guys let's multiply 4 times 7 that is 28 32 7 times 1 and 7 times 1 7 plus 2 is 9. in this opinion guys this is 98 your a sub 3 is 90 that is the answer for item number 2. now let's go with item number three number three when you encounter some fractions we have three blank blank and one over nine don't be afraid about that okay using the formula a sub n is equal to a sub 1 times r raised to n minus 1 your a sub n is equal to 1 over 9 as our last term a sub 1 is equal to 3 your n is equal to 1 2 and 3 and 4. so same problem plug in for the formula a sub n is 1 over 9 1 over 9. is equal to three first term your r is unknown your number of terms is four minus one so we have one over nine is equal to three r cube divide both sides by three okay when you're dividing a fraction by a whole number my whole number it will move to multiplication it will become one over nine times you'll receive probably on three and that is one third so one over nine times one over three and that will give you 1 over 27 so this is 1 over 27 cancel nothing your r is our cube okay so we are the square root of get the cube root of 1 over 27 and get the cube root of r cubed okay sodium the cube root of 1 over 27 is 1 over 3. and i don't know when this is your r it is a b and your common ratio is equal to r and we will use the common ratio to find the second and third term so for the second term a sub two that is three times one third so on three times one type button is the same as three over three 3 your a sub 2 is equal to 1. this is the second third now for the third term [Music] which is one times one third one times one is one another invisible one that will give us the answer of a sub three [Music] so if you're new to my channel don't forget to like and subscribe and hit learning about button for you to be updated in latest uploads again it's me teacher going bye