 n minus 1 and an these were n terms in a sequence and what type of sequence was it there is some important property which we discussed last time and this was that if a2 upon a1 is equal to a3 upon a2 is equal to a4 upon a3 what all these what all are these these are you know nothing but they are a ratio of consecutive terms so a n upon a n minus 1 if it is a constant value r okay so then we say that this particular sequence is geometric progression okay this is GP hmm so very vital for all your exams especially next year's KPI and yes if you're writing NTSC this is going to be very very helpful geometric progression right so then we had this first thing of general term general term of GP okay so you can see a2 by a1 is equal to r this implies a2 is equal to a1 r then a3 upon a2 was r again so this implies a3 is equal to a2 times r and a2 itself was a1 r so a1 r square correct and so on and so forth if you see a4 by a3 is equal to r this implies a4 is equal to a3 r which is equal to again a1 r cubed so what do we say we say keep on going like that a to the a n nf term of a GP will be given by a1 r and this is the first formula and corresponding formula in ap was nf term a n for tn was a plus n minus 1 b here you can say even let's say a is equal to a1 first term first term then any nf term of the GP will be of the form of a r n minus 1 so corresponding formula for nf term of a GP this was corresponding this was formula for nf term of an ap where d is the common difference here r is the common ratio any difficulty in this clearly a GP with zero is not allowed right so none of a1 a2 a3 can be 0 why because then division by 0 is not allowed so a1 a2 a3 none of them are equal to 0 so 0 is not allowed in GP r could be plus minus r could be greater than 1 r could be less than 1 whatever value but not 0 okay find question number one a9 for far no not necessary not necessary natural numbers not necessary rational numbers we are mostly dealing with rational numbers so here is the example which will make it clear 1 by 4 minus 1 by 2 1 minus 2 find ninth term in this quick so this is a GP first of all tell me the common ratio find the ninth term find the ninth term quickly n is less than 0 where is n less than 0 and is number of number of if n is natural see number cannot be number of number of something cannot be anything but natural number I'm talking about the the elements the members of GP could be anything there's no restriction as to it has to be a natural number but number of terms has to be natural numbers only no so a9 is how much you cannot have 22.5th term no significance of such thing so yes quick find out a9 tell me what is the first term in this use the formula use the formula a n is a r and minus 1 clearly a is 1 upon 4 how do I find out find out r take the ratio of any two consecutive terms so minus 1 by 2 divided by 1 by 4 this is r this is minus 2 isn't it so what is a9 is equal to a r to the power 8 and minus 1 so a is 1 upon 4 into minus 2 to the power 8 okay so answer is 64 question number 2 again keep in mind a n is equal to a r and minus 1 now consider the GP to 1 1 by 2 1 by 4 right so in this GP which term is 1 1 upon 128 find n in this case right so a is 2 r is how much is r 1 upon 2 correct so a n is equal to 1 by 128 is equal to a into r n minus 1 you have to find out n minus 1 so 120 is 2 to the power minus 7 1 upon 128 and this is a is 2 and r is 1 by 2 to the power n minus 1 correct so 2 to the power minus 7 is equal to 2 to the power 1 into 2 to the power minus n minus 1 right so this is 2 to the power minus 7 is equal to 2 to the power of 1 minus n minus that means minus 7 is equal to 1 minus n plus 1 right that means n is equal to 9