Exercise n°1Identify the pattern of the following sequence.1, 1, 2, 3, 5, 8, 13, . ..Exercise n°2Write out the first four items of the sequences whose general terms are:a. 𝑎𝑎 𝑛𝑛 = 2𝑛𝑛 + 1b. 𝑎𝑎 𝑛𝑛 = 3𝑛𝑛+1c. 𝑎𝑎 𝑛𝑛 = 𝑛𝑛+1𝑛𝑛d. 𝑎𝑎 𝑛𝑛 = 1𝑛𝑛 2Exercise n°3What is the limit of the following sequences, and determine whether they converge or diverge?a. 𝑎𝑎 𝑛𝑛 = 2𝑛𝑛𝑛𝑛+1b. 𝑎𝑎 𝑛𝑛 = 3 + (−1)𝑛𝑛c. 𝑎𝑎 𝑛𝑛 = 2𝑛𝑛2𝑛𝑛 −1d. 𝑎𝑎 𝑛𝑛 = (−1)𝑛𝑛𝑛𝑛!e. 𝑎𝑎 𝑛𝑛 = 5𝑛𝑛+73𝑛𝑛−5f. 𝑎𝑎 𝑛𝑛 = 𝑛𝑛 2 +12𝑛𝑛−3Exercise n°4Identify the type of the following sequences.a. 𝑎𝑎 𝑛𝑛 = 2𝑛𝑛 + 3b. 𝑎𝑎 𝑛𝑛 = 2𝑛𝑛Exercise n°5Generate the general term of the following sequence.− 11 , 32 , − 76 , 1524 , − 31120 , …Exercise n°6Find the general term of the following sequence.Given the function 𝑓𝑓(𝑥𝑥) = 𝑒𝑒 𝑥𝑥3 𝑎𝑎 𝑛𝑛 = 𝑓𝑓(𝑛𝑛−1) (0)Exercise n°7Applying l’Hôpital’s rule, evaluate lim𝑛𝑛→∞(𝑛𝑛+1)𝑒𝑒 𝑛𝑛Exercise n°8Applying the squeeze theorem, determine the convergence or divergence of the following sequences:a. 𝑎𝑎 𝑛𝑛 = sin 𝑛𝑛𝑛𝑛 2b. 𝑎𝑎 𝑛𝑛 = (−1)𝑛𝑛 ∙ 1𝑛𝑛c. 𝑎𝑎 𝑛𝑛 = 𝑛𝑛!𝑛𝑛 𝑛𝑛Exercise n°9Investigate whether the following sequences are increasing, decreasing or neither.a. 𝑎𝑎 𝑛𝑛 = 𝑛𝑛𝑛𝑛+1b. 𝑎𝑎 𝑛𝑛 = 𝑛𝑛!𝑒𝑒 𝑛𝑛 , for 𝑛𝑛 ≥ 2Exercise n°10Show that the following sequence is bounded.𝑎𝑎 𝑛𝑛 = 3 − 4𝑛𝑛 2𝑛𝑛 2 + 1

. Prove, using the ϵ − N definition oflimit, that the sequence (an) given byan = n2n − 1converges to 12 .2. Prove using the definition of limit thatthe sequence (an) given byan = 1(3n − 1)converges to 0.3. Evaluate the following limits(a) limn→∞n4 + 3n1 + n3 + n5 .(b) limn→∞3n − 13n − 3n−1(c) limn→∞√n + 1 − √n(d) limn→∞ nr4 − 1n − 2!4. Let (an) be a sequence. Show that iflimn→∞ an exists, then (an) is bounded.5. Show that the sequence (an) given byan = n2n + 1does not converge.6. Use the sandwich theorem to findlimn→∞1(3n − 1)7. (a) Let n ∈ N. Define functions f, gbyf (x) = (1 + x)nandg(x) = 1 + nxShow by induction that for all nat-ural numbers n ≥ 1 and real num-bers x ≥ −1f (x) ≥ g(x).(b) Sketch graphs f and g case for thecase n = 3.(c) Let a > 1. Show that the sequence(an) is unbounded above.(d) If |a| < 1, show that limn→∞ an = 08. Let x ∈ R. Define a sequence of partialsums by (sn) bys1 = 1s2 = 1 + xs3 = 1 + x + x2... = ...sn = 1 + x + x2 · · · + xn−1Show that (sn) converges if and only if|x| < 1. Find the limit of the sequence(sn) if x = 910 .9. Evaluate:(a) limn→∞1√n(b) limn→∞5n + (−1)n4n(c) limn→∞n34n(d) limn→∞√n√n − 2(e) limn→∞sin n + n3n3(f) limn→∞(−1)n√n1https://pennance.us1

3.1 If and are the first three terms of a linear sequence,2𝑥 − 1, 𝑥 + 5 4𝑥 − 1determine the following:3.1.1 the value of 𝑥. (2)3.1.2 the first three terms. (1)3.1.3 the general rule of the term. (1)3.2 Given the quadratic pattern: 0, − 3, − 10, − 21, − 36...3.2.1 Determine the formula for the term of the pattern.𝑛𝑡ℎ (6)3.2.2 Which term is equal − 820? (5)[15

Instructions: Given one form of a sequence, write the other form. Recursivean=an−1⋅−3𝑎𝑛=𝑎𝑛−1⋅−3a1=2𝑎1=2Explicit Forman=𝑎𝑛= Answer 1 Question 7 (( Answer 2 Question 7 )n−1

Prove from first principles that the sequence (an) is convergent, wherean = (−1)n + n2n + 1for all n.

Prove from first principles that the sequence (an) is convergent, wherean = (−1)n + n2n + 1for all n. [5 marks](b) Suppose that a sequence (cn) satisfies |cn| > C|cn−1| for all n ≥ 1, whereC > 1. Show that (|cn|) → ∞. Does it follow that either (cn) → ∞ or(cn) → −∞? Justify your answer. [6 marks](c) If |dn| > |dn−1| for all n, does it follow that (|dn|) → ∞? Justify youranswer with a proof or a counterexample.