Question

3. Prove by induction that∑_(r=1)^n▒〖r(r+4)=1/6 n(n+1)(2n+13)〗.

5. It is given that u_1=1 and u_(n+1)=3u_n+2n-2 where n is a positive integer.
Prove, by induction, that u_n=3^n/2-n+1/2.

14. The rth term of a sequence is given by u_r=r(3r+1),r=1,2,3,…
i) Write down the values of ∑_(r=1)^n▒u_r for n=1,2,3, and 4.
ii) Make a conjecture for a formula for ∑_(r=1)^n▒u_r , giving your answer in the form nf(n), where f(n) is a function of n.
iii) Prove by induction a formula for ∑_(r=1)^n▒u_r .


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