Review practice · Section 2 of 4

Given $\sum_{r=2}^{n-1}[2r+4(1/(r+1)-1/r)]=n^2-n+4/n-4$, show

Given $A_n=\sum_{r=1}^n1/[r(r+2)]=3/4-1/[2(n+1)]-1/[2(n+2)]$:

(i) Deduce $A_n<3/4$ for all positive integers $n$.1 mark

(ii) Given also $\sum_{r=1}^n1/[r(r+1)]=1-1/(n+1)$, find $\sum_{r=1}^n1/[2r(2r+2)]$.2 marks

(iii) Use these results to find $\sum_{r=1}^n1/(4r^2-1)$ in terms of $n$.4 marks

Given $\sum_{r=1}^n1/(\sqrt{r+1}+\sqrt r)=\sqrt{n+1}-1$, deduce $\sum_{r=1}^n1/\sqrt r>2(\sqrt{n+1}-1)$.2 marks

The n-th row from the top of a board has n tiles, labelled $1,2,\ldots,n$ from left to right, as in the diagram. Given $\sum_{r=1}^nr^2=n(n+1)(2n+1)/6$, find the sum in the k-th row and hence show the sum of all numbers in n rows is $n(n+1)(n+2)/6$.4 marks

The sequence begins $1$, $(1+2)$, $(1+2+2^2)$, $(1+2+2^2+2^3)$, … .

(i) Show its n-th term is $2^n-1$.2 marks

(ii) Find the sum of its first n terms.3 marks

Find the first seven terms of $u_{n+1}=(n+1)u_n$, $u_1=1$, and a formula for $u_n$.