General And Middle Terms

Step 1: General term in the sequence. The general form of the nth term in the sequence is the product of three consecutive odd numbers, which is given by: Step 2: Relating the sum to the given expression. We are given that: Thus, we equate the sum to the expression .
Step 3: Solving for . Substituting into the equation: Solving for : Thus, .

Step 1: Binomial Expansion. The expansion of is given by: Step 2: Equidistant Terms. The ratio of the equidistant terms from the middle term is given as .
From this, we deduce that the sum of all terms is .

We are given that there are 6 different novels and 3 different poetry books. We need to select 4 novels and 1 poetry book, and arrange them in a row on a shelf with the condition that the poetry book is always in the middle.
Step 1: Arranging the poetry book in the middle Since the poetry book must always be in the middle, we have only 1 choice for the position of the poetry book. There is only 1 position for the poetry book in the middle of the 5 positions on the shelf.
Step 2: Selecting and arranging the novels We need to select 4 novels from the 6 available novels. The number of ways to choose 4 novels from 6 is given by the combination formula: After selecting the 4 novels, we can arrange them in the 4 remaining positions. The number of ways to arrange 4 novels is , which is: Step 3: Selecting the poetry book Since there are 3 different poetry books, we can choose any 1 of them in 3 ways.
Step 4: Calculating the total number of arrangements The total number of arrangements is given by: Thus, the total number of possible arrangements is .