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MATH 3210 - SUMMER 2008 - ASSIGNMENT
Induction and ∑^ notation
(1) Expand the following expressions i.e. write the first 3 terms and last three terms (with dots in between) in each sum: (a)
∑^ n i=
i
(b) ∑^ n i=
(n − i)
(c) ∑^ n i=
2 i
(d)
n∑+ k=
2 k−^1
(e) ∑^ n i=
i
(f)
n∑− 2 j=− 2
3 · 2 j
(2) Write the following expressions in ∑^ form: (a) 1 + 3 + 5 + · · · + 13 (b) 6 + 9 + 12 + · · · + 24 (c) 1 + 3 + 9 + 27 + 81 (3) Find and prove formulas for the following expressions:
(a)
∑^ n i=
i −
∑^ n+ j=
j =
(b) ∑^2 n i=
(− 3 i^2 − 2 i) − ∑^2 n j=n
(− 3 j^2 − 2 j)
(c)
∑^ n k=
k −
∑^ n k=
k + 1 (d) ∑^ n k=
k(k + 1) (hint: use the previous item). (4) Prove the following statements using induction (can you prove them in another way?) (a) If 0 < a < b then for all n ∈ N: an^ < bn 1
(b) Prove that ∑^ n i=
2 i^ = 2n+1^ − 2
(c) Prove that ∑^ n k=
(7 + 3(k − 1)) =^32 n^2 +^112 n
(d) Prove that for any real a, q and for any integer n: ∑^ n i=
aqi^ = a^ −^ aq
n+ 1 − q (5) (a) Using the formula for (nk^ )^ prove that: (nk^ )^ = (^ n−nk^ ) (b) Prove the same thing but only using the definition of (nk^ )^ (i.e. that it is the number of k-element subsets of the set { 1 ,... , n}). (6) (a) Using the formula for (nk^ )^ prove that: (n+1 k^ )^ = (nk^ )^ + (^ k−n 1 ) (b) Prove the same thing but only using the definition of (nk^ )^ (i.e. that it is the number of k-element subsets of the set { 1 ,... , n}). (7) Using induction prove the binomial formula (hint: if you get stuck you can use the proof in page 12 of the text but you must explain every step).
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