 Hello and welcome to the session. In this session we discuss the following question which says if the seventh of an HP that is harmonic progression are respectively 2 upon 5 and 2 upon 7, find the first term of the HP. Before we move on to the solution, let's see when a sequence say a sequence A, B, C and split progression. So this sequence is a harmonic progression that is HP when the reciprocals of these terms of the sequence that is 1 upon A, 1 upon B, 1 upon C which is AP, 1 the whole into B where this A is into the solution which is the first term of the HP where given a progression if we equal to a sequence is said to be harmonic progression when the reciprocals of the terms form an automatic progression. So from here we get would be equal to 5 upon 2 that is the reciprocals of the, now we remember Nth term of an AP is equal to A plus N minus 1 into D. Therefore we have A1 which we have supposed to be the first term of the AP minus 1 which is 6 into D is equal to 5 upon 2. Let this be result 1. Now next we are also given HP is equal to upon 7 which means the AP would be its reciprocals which is 7 upon 2. So therefore we have A1 which is the first term of the and that is 9 into D is equal to 7 upon 2. Let this be result to solve these two equations we get the values of A1 and D. So equation 1 from equation 2 we get A1 plus 9 D minus A1 D is equal to 7 upon 2 minus 5 upon. So further we have A1 cancels with minus A1 9 D minus 3D is equal to 2 upon 2 which is 1. This means 3D equal to 1 or you can say D is equal to 1 upon 3. Let me find out the value of A1. Next is the value of D into D which is 1 upon 3 is equal to 7 upon 10 is 9. So there we have A1 plus 3 is equal to 7 upon 2. 1 is equal to 7 upon 2 minus 3 which gives us A1 equal to 7 minus 6 upon 2 that is 1 upon 3 AP is equal to 1 upon 2 and this means HP would be the reciproc 1 upon 2 which would be equal to 2. The first term of the HP to be A hope you understood the solution of this question.