 Hello and welcome to the session. In this session we will discuss geometric progression or GP. A sequence is said to be a geometric progression or a GP if the ratio of any term to its preceding term is same throughout. That is a sequence a1, a2, a3 and so on up to an is called geometric progression or a GP if each term is non-zero ak plus 1 upon ak is equal to r that is a constant for some k greater than equal to 1. Consider geometric progression a, a r, a r square, a r cube and so on. Here a is the first term of the GP, r is the common ratio of the GP. Then the general term or the nth term of a GP is given as an equal to a into r to the power n minus 1 where again we know that a is the first term, r is the common ratio and n would be the number of terms of the GP. A GP with finite terms is finite and GP with infinite number of terms is infinite GP. Now the sum of first n terms of GP given by Sn is equal to a into 1 minus r to the power n upon 1 minus r or this could be also equal to a into r to the power n minus 1 upon r minus 1. This is the sum of first n terms of a GP. This is the case when r is not equal to 1. If in case we have r is equal to 1 then the sum of first n terms that is Sn would be equal to n into a. Consider this GP here the first term a is equal to 1. The common ratio r is equal to 4 and the number of terms of the GP given by n is equal to 5. Let's find out the sum of these 5 terms of the GP that is we need to find S5 which is equal to a that is 1 into r raised to the power n that is 4 to the power 5 minus 1 upon r minus 1 which comes out to be equal to 341. That is sum of the first 5 terms of the given GP is 341. Next we shall discuss geometric mean or GM. If we are given two positive numbers a and b then the geometric mean or GM of the numbers a and b is square root ab. Given any two numbers a and b we can insert as many terms or as many numbers between them and the resulting sequence would be a GP. Like suppose we insert the terms g1, g2 and so on up to gn between the numbers a and b then this resulting sequence would be a geometric progression or a GP. Now the total number of terms of this GP would be n plus 2 since we have n numbers inserted between the two numbers a and b so this n plus the two numbers a and b. Total becomes n plus 2 terms. Now here the last number of the GP b is equal to a into r to the power n that is the total number of terms in this case it would be n plus 2 minus 1 which becomes equal to a into r to the power n plus 1. This gives us r equal to b upon a to the power 1 upon n plus 1. Now g1 would be equal to a into r on substituting the value of this r we get g1 is equal to a into b upon a to the power 1 upon n plus 1 then we have g2 is equal to a into r to the power 2. So this becomes equal to a into b upon a whole to the power 2 upon n plus 1 and so like this we get the different numbers that we have inserted between the numbers a and b. The term gn would be given by a into r to the power n so that is equal to a into b upon a whole to the power n upon n plus 1. So this is how we find the terms inserted between any two given numbers. Suppose we have two numbers given to us 3 and 81 and we need to insert two numbers g1, g2 between these two numbers so that the resulting sequence is a GP. Here we have a is equal to 3 b is equal to a d1 and n is equal to 4. So here b that is 81 is equal to a into r to the power n minus 1. So we get r2 is equal to 81 upon 3 that is 27 and thus we get r equal to 3. Now we have got r equal to 3 so g1 is equal to a into r which is equal to 3 into 3 equal to 9 then g2 is equal to a into r to the power 2. This becomes equal to 3 into 3 to the power 2 that is equal to 27. So we get the sequence as 392781 which is a GP. So this is how we insert any number of terms between any two given numbers to get the resulting sequence a GP. Now let's discuss the relationship between arithmetic mean am and geometric mean gn. Given any two positive numbers a and b we know that the arithmetic mean am of the two numbers a and b given by a is equal to a plus b upon 2 and the geometric mean gm of the two numbers a and b denoted by g is equal to square root ab. Arithmetic mean a minus geometric mean g is equal to a plus b upon 2 minus square root ab. This is further equal to root a minus root b the whole square upon 2. This is obviously greater than equal to 0 and thus we get that a is greater than equal to g that is arithmetic mean is greater than equal to the geometric mean. Now let's discuss some two n terms of special series. First let's find out the sum of first n natural numbers. Let it be denoted by Sn that is Sn is equal to 1 plus 2 plus 3 and so on up to n. So this sum Sn is equal to n into n plus 1 upon 2. Next is sum of squares of first n natural numbers. Let this be denoted by Sn which is equal to 1 square plus 2 square plus 3 square plus and so on up to n square. So this Sn would be equal to n into n plus 1 into 2 n plus 1 upon 6. Next is sum of cubes of first n natural numbers. Let it be denoted by Sn equal to 1 cube plus 2 cube plus 3 cube plus and so on up to n cube. Then this Sn is equal to n into n plus 1 the whole square upon 4. Let's try and find out the sum to n terms of the series whose nth term is given by that is we have An is equal to 2n minus 1 the whole square. Consider the kth term of the series given by Ak equal to 2k minus 1 the whole square which is equal to 4k square minus 4k plus 1. Since we need to find the sum to n terms so we have summation Ak k goes from 1 to n is equal to summation k goes from 1 to n 4k square minus 4k plus 1. This is further equal to 4 into summation k goes from 1 to n k square minus 4 into summation k goes from 1 to n k plus summation k goes from 1 to n 1. This is equal to 4 into now we know that the sum of the squares of first n natural numbers is given by n into n plus 1 into 2n plus 1 upon 6 minus 4 into now. Submission k goes from 1 to n is the sum of the first n natural numbers given by n into n plus 1 upon 2 plus n. This comes out to be equal to n upon 3 into 2n plus 1 into 2n minus 1. So, this is the sum of the n terms of the series whose nth term is given by this n. This completes the session hope you have understood the concept of geometric progression relationship between arithmetic mean and geometric mean and the sum of n terms of some special series.