 Hello and welcome to the session. In this session, we will discuss about harmonic progression. Let us first start with the definition. The sequence formed by the reciprocal of the terms of an arithmetic progression is called a harmonic progression. It is denoted by h p. Consider the following series. First, 1 plus 1 by 2 plus 1 by 3 plus 1 by 4 plus and so on. Second, 2 by 3 plus 2 by 5 plus 2 by 7 plus 2 by 9 plus and so on. First, 1 by a plus 1 by a plus d plus 1 by a plus twice of d plus 1 by a plus twice of d plus and so on. These series are in harmonic progression. Let us see how to get the reciprocal of the terms of these series. The first series becomes 1 plus 2 plus 3 plus 4 plus and so on. The second series becomes 3 by 2 plus 5 by 2 plus 7 by 2 plus 9 by 2 plus and so on. And the third series becomes a plus a plus d plus a plus twice of d plus a plus twice of d plus and so on. Now, we can see that all these series are in arithmetic progression which is denoted by a p. Thus, by the given definition, the given series are in harmonic progression. Now, we will find out how to calculate the L.S. term of a harmonic progression. The L.S. term of a harmonic progression is the reciprocal of the L.S. term of the arithmetic progression found by the reciprocal of the terms of harmonic progression. Therefore, the L.S. term of the harmonic progression that is 1 by a 1 by a plus d 1 by a plus twice of d 1 by a plus twice of d and so on is given by 1 upon a plus n minus 1 into d. Let us now discuss what a harmonic mean is. Harmonic mean is denoted by capital H. In a harmonic progression or a sequence, each term is the harmonic mean of the neighboring terms let A and C be the given numbers and FBH harmonic mean between them since C are in harmonic progression. It implies that 1 by a 1 by H 1 by C are in arithmetic progression. We have earlier stated that in an arithmetic progression twice the middle term is equal to the sum of the preceding and the following term. So, we have 1 by a plus 1 by C is equal to 2 by H which implies that a plus C by a into C is equal to 2 by H. Thus, harmonic mean H is given by twice of a into C upon a plus C. So, H is the harmonic mean. We shall now discuss about arithmetic geometric series. A series obtained on multiplying the corresponding terms of an arithmetic progression and geometric progression is called an arithmetic geometric series. The general or standard form of such a series is given by a plus a plus d into r plus a plus twice of d into r square plus a plus twice of d into r cube plus and so on up to a plus n minus 1 into d So, multiplied by r raised to power n minus 1 plus and so on each term of this series is found by multiplying the corresponding terms of two series that is a plus a plus d plus a plus twice of d plus a plus twice of d plus and so on up to a plus n minus 1 into d plus and so on which is an arithmetic progression and 1 plus r plus r square plus r cube plus and so on up to r raised to power n minus 1 plus and so on which is an geometric progression. Now, let's find the sum of an arithmetic geometric series. Let S n be the sum of n terms of the series then S n is equal to a plus a plus d into r plus a plus twice of d into r square plus and so on up to a plus n minus 1 into d multiplied by r raised to power n minus 1 and Markov's equation as 1. Now, multiply the whole equation by r on both the sides and we get r into S n is equal to a into r plus a plus d into r square plus and so on up to a plus n minus 2 into d multiplied by r raised to power n minus 1 plus a plus n minus 1 into d multiplied by r raised to power n and Markov's equation as 2 on subtracting equation 1 from equation 2 we get 1 minus r into S n is equal to a plus d into r plus d into r square plus and so on up to d into r raised to power n minus 1 minus of a plus n minus 1 into d multiplied by r raised to power n and is equal to a plus d into r into 1 minus r raised to power n minus 1 whole upon 1 minus r minus of a plus n minus 1 into d whole multiplied by r raised to power n thus S n is equal to a upon 1 minus r plus d into r into 1 minus r raised to power n minus 1 whole upon 1 minus r d raised to power n minus of a plus n minus 1 into d whole multiplied by r raised to power n whole upon 1 minus r now if we have the value of r less than 1 then r raised to power n minus 1 tends to 0 and r raised to power n tends to 0 as n tends to infinity thus if n is infinity in the above equation we get S infinity is equal to a upon 1 minus r plus d into r upon 1 minus r d whole square let us see what method of differences is in certain series which are neither arithmetic progression nor geometric progression it may happen that the successive differences like t2 minus t1, t3 minus t2, t4 minus t3 etc may form an arithmetic progression or geometric progression in that case the element term can be found then the sum can be deduced for example consider the series 1 plus 2 plus 3 plus 5 plus and so on let tn be the element term and S be the sum of first n terms then S n is equal to 1 plus 2 plus 3 plus 5 plus and so on up to tn minus 2 plus tn minus 1 plus tn when S is equal to 1 plus 2 plus 3 plus 5 and so on up to tn minus 2 plus tn minus 1 plus tn now subtract the two equations we have S minus S that is 0 is equal to 1 plus 1 plus 1 plus 2 plus 3 plus and so on up to n minus 1 terms minus S tn which implies that tn is equal to 2 plus 1 plus 2 plus 3 plus and so on up to n minus 1 terms now here we can see that an arithmetic progression is found so we have tn is equal to 2 plus now taking the sum of the arithmetic progression we get n minus 1 by 2 into 2 into 1 plus n minus 1 minus 1 multiplied by 1 so this is equal to 2 plus n minus 1 by 2 into 2 into 1 that is 2 plus n minus 1 minus 1 into 1 which is equal to n minus 1 minus 1 that is n minus 2 which is equal to 2 plus n minus 1 by 2 into n that is 2 plus n square minus n by 2 and taking the LCM we have 4 plus n square minus n by 2 so tn is given by n square minus n plus 4 multiplied by 2 this complete our question for you to enjoy this question