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- PROVE 1.11+2.2! +3.3!+----n.n!=(n++)!-1 A n>=1I've looked at the existing answer to this question already but it doesn't quite make sense to me. First, when I plug in a random positive integer for N, the formula given as the answer to this problem doesn't produce the same outcome as when I plug the summation into an online calculator. Second, I'm having a hard time understanding how we can limit the answer formula to a set number of parts for all possible positive integers of N. Is it not that for all N, the inner portion of the by parts equation 1/N[f(n)-f(n+N)] when put into summation would have all values from f(1) to f(N) and also have N number of remaining portions at the end uncanceled? How can this be simplified to work for any N? I think I have a difficult time reasoning with infinity so perhaps that's where the issue lies. Any help would be greatly appreciated, Thank You!4 Suppose we have 3000 identical sheets separated into packs of 25 sheets each. Use generating functions to calculate the number of ways in which you can distribute the packs of 25 sheets among four groups of students so that each group has at least 150 sheets and more than 1000 sheets. Express your answer using binomial coefficients and include the calculations made. Explain your solution in detail.
- Justify recursion for 2nd kind Stirling numbers:A model for the number of lobsters caught per year is based on the assumption that the number of lobsters caught in a year is the average of the number caught in the two previous years. 1. Find a recurrence relation for {Ln}, where Ln is the number of lobsters caught in year n, under the assumption for this model. 2. Find Ln if 100,000 lobsters were caught in year 1 and 300,000 were caught in year 2.How can the sum of k * (n choose k) = n(2 to the power of n-1) be proven akgebraically ?
- A man finished a job in 5 days. On the first day, he finished 1/m of the job. On the second day, he finished 1/n of the job left. On the third day, he finished 1/m of the job left, and on the fourth day, 1/n of the job left. On the last day, he finished 1/4 of the remaining job. find m and n.Prove it by mathematical induction. PLease don't provide an internet solution as i didn't get to understand from that.Find the power series solution for the equationy''-y=xProvide the recurrence relation for the coefficients and derive at least 3 non-zero terms of the solution. I just have no clue how to incorporate the =x as most power series I have worked with just =0