 Hello and welcome to the session. I am Aasha and I am going to help you with the following question which says the first term of a GP is 1. The sum of third term and fifth term is 90. Find the common ratio of GP. Let us now begin with the solution. And here we are given that the first term it is denoted that A is equal to 1 and the sum of equal to 90. So let the third term be denoted by A3 plus A5 is equal to 90. And now let the common ratio be equal to r which is greater than 1. So the third term will be AR square and the fifth term will be A into r raised to the power 4. Since the GP series the first term A and common ratio r will be of the form A into AR into AR square plus AR cube and so on. Now this is the first term A, AR this is the second term AR square is the third term and we are given that sum of third and fifth term is 90. So we must have A3 plus A5 is equal to AR square plus AR raised to the power 4 is equal to 90. Or we can say that AR square plus AR raised to the power 4 is equal to 90. Now A is 1 so on substituting the values we have AR raised to the power 4 plus AR square minus 90 is equal to 0. Now let r square is equal to x so this implies AR raised to the power 4 is equal to x square. So let this be equation number 1. So 1 can be written as x square plus x minus 90 is equal to 0. Now by splitting the middle term this can further be written as x square plus 10x minus 9x minus 90 is equal to 0. Now taking x common from the first two terms and minus 9 common from the last two terms this can further be written as x into x plus 10 minus 9 into x plus 10 is equal to 0 or x plus 10 into x minus 9 is equal to 0. Now if the product of two numbers is equal to 0 then at least one of them is 0 so this implies either x plus 10 is equal to 0 or x is equal to 9. So this further implies that x is equal to minus 10 or x is equal to 9 that implies that r square is equal to minus 10 or r square is equal to 9. Now the negative value is rejected since it will preserve to the complex number as a ratio. So we have r square is equal to 9 or r is equal to plus minus root over 9 which gives plus minus 3 as the answer is the common ratio of GPS plus minus 3. So this completes the solution. Take care and have a good day.