Module 7 Homework Problems

Write out the sequences generated by the following rules. Be sure to list at least 6 terms!

1. \(A=(a_i)_{i=1}^\infty\) where \(a_i=i^2\)
2. \(B=(b_i)_{i=1}^\infty\) where \(b_i=i^3\)
3. \(D=(d_i)_{i=1}^\infty\) where \(d_i=2^i\)
4. \(C=(c_i)_{i=1}^\infty\) where \(c_i=\frac{1}{i}\)
5. \(D=(d_i)_{i=1}^\infty\) where \(d_i=(-1)^i\cdot i\)
6. \(E=(e_i)_{i=1}^\infty\) where \(e_i=(-1)^{i+1}\cdot 2^i\)

Find a closed formula (rule) that generates the following sequences. Note that the sequences you generated in previous problems may help!

7. \(A=(3,6, 11, 18, 27, …)\)
8. \(B=\{\frac{2}{1},\frac{4}{2},\frac{8}{3},\frac{16}{4},\frac{32}{5},…\}\)
9. \(D=\{-1,1,-1,1,-1,1,-1,1,-1,….\}\)
10. \(E=\{\frac{1}{2},-\frac{1}{4},\frac{1}{6},-\frac{1}{8},\frac{1}{10},…\}\)

Find the closed formula generating the following arithmetic sequences.

11. \(N=\{1,2,3,4,5,…\}\)
12. \(M=\{-8,-3,2,7,12,…\}\)
13. \(P=\{4,2,0,-2,-4,-6,-8,…\}\) (Never said arithmetic sequences needed to be increasing! Same principles apply. Business as usual. You’re just adding negative numbers.)

Find the sum of the terminating arithmetic sequences given below.

14. \(K=\{3,6,9,12,15,…, 303\}\) (Hint: be careful with the number of terms being added!)
15. \(L=\{5,1,-3,-7,-11,…, -235\}\) (Hint: Don’t let weird circumstances scare you! See if we can carry on as usual anyway!)

Find the closed formula generating the following geometric sequences

16. \(A=\{5,35, 245,1715,…\}\)
17. \(B=\{\frac{4}{9},\frac{8}{27},\frac{16}{81},\frac{32}{243},…\}\)

Find the sum of all entries of the geometric sequences given below.

18. \(F=\{4,16,64,256,…, 1048576\}\)
19. \(G=\{\frac{5}{3},\frac{5}{9},\frac{5}{27},\frac{5}{81},…,\frac{5}{4782969}\}\)
20. \(H=\{\frac{1}{3},\frac{1}{9},\frac{1}{27},\frac{1}{81},…\}\) (Yes, we are dealing with an infinite sequence here. Whatever… Business as usual (use the same approach as before)).
21. (Bonus +10) Note that \(0.999999\overline{9}=\frac{9}{10}+\frac{9}{100}+\frac{9}{1000}+\frac{9}{10000}+…\) (This is because each term of the sum gives you each decimal digit; i.e. \(\frac{9}{10}=0.9, \frac{9}{100}=0.09, \frac{9}{1000}=0.009\), etc. So, when you add them all up, you will get \(0.9999\overline{9}\).) Use this expansion (and techniques for finding sums of geometric sequences) to show/prove that \(0.999999\overline{9}=1\).

Calculate the following sums.

22. \(\sum_{i=1}^{10} 3i^2\)
23. \(\sum_{i=1}^{10} i\)
24. \(\sum_{i=4}^{15} 2\) (No, this is not a typo)
25. \(\sum_{i=1}^n 5\), where \(n\) is an arbitrary natural number. (Helps to write out the sum as usual)
26. \(\sum_{i=1}^8 (-1)^i\)
27. (Bonus +10) Prove that “If \(n\) is even, then \(\sum_{i=1}^n (-1)^i=0\)” (Hint: A sufficiently good explanation based on grouping \(1\)’s and \(-1)’s would be a good foundation for a proof. As with all proofs, don’t ask “am I allowed to do that?” but instead ask “does it make sense for me to say that? Is what I said true?”)

Calculate the value of the following products

28. \(\Pi^5_{i=1} \frac{2}{3}\)
29. \(\Pi^6_{i=3} (3i-4)\)
30. \(\Pi^{1000}_{i=1} 2^i \cdot \Pi^{1000}_{i=1} \left(\frac{1}{2}\right)^i\)
31. (Bonus +5) \(\Pi^{25}_{i=1} 3i \cdot \Pi^{25}_{i=5} \frac{1}{3i}\) (note that the starting indices are different. You’ll need to think carefully about how to compute this using product properties. No bonus points will be given for direct calculation of a massive 50-factor product)