 Hi, I'm Kanika and I'm going to help you to solve the following question. The question says evaluate summation k goes from 1 to 11, 2 plus 3 to the power k. Let's now begin with the solution. In this question we have to evaluate summation k goes from 1 to 11, 2 plus 3 to the power k. Now we will open this summation. So this is equal to 2 plus 3 to the power 1 plus 2 plus 3 to the power 2 plus 2 plus 3 to the power 3 plus 2 plus 3 to the power 4 plus so on till 2 plus 3 to the power in limit. Now this expression is equal to 2 plus 2 plus 2 so on till 11 times plus 3 to the power 1 plus 3 to the power 2 plus so on till 3 to the power 11. Since 2 appears 11 times so 2 plus 2 plus 2 till 11 times is equal to 22 times the sum of b series. We know that sum of n terms of gp that is Sn is equal to a into r to the power n minus 1 upon r minus 1 if r is greater than 1. Now here the first term that is a is equal to 3 and r is also equal to 3 since on dividing 3 to the power 2 by 3 to the power 1 we get 3 and number of terms is 11. So the sum of these series is equal to 3 into r to the power n which is 3 to the power 11 minus 1 upon 3 minus 1. Now this is equal to 22 plus 3 into 3 to the power 11 minus 1 upon 2. Now this can further be written as 22 plus 3 by 2 into 3 to the power 11 minus 1 hence a required answer is 22 plus 3 by 2 into 3 to the power 11 minus 1. This completes the session bye and take care.