MA3A9. Students will use sequences and series
... The launching activity begins by revisiting ideas of arithmetic sequences studied in eighth and ninth grades. Definitions, as well as the explicit and recursive forms of arithmetic sequences are reviewed. The task then introduces summations, including notation and operations with summations, and sum ...
... The launching activity begins by revisiting ideas of arithmetic sequences studied in eighth and ninth grades. Definitions, as well as the explicit and recursive forms of arithmetic sequences are reviewed. The task then introduces summations, including notation and operations with summations, and sum ...
Contents
... toppings, how many different pies would they need to prepare to guarantee that they have any combination ready that someone might ask for? You might ask a couple of questions. First, can you get double of one topping? That obviously changes the answer. Second, does the order of the toppings matter? ...
... toppings, how many different pies would they need to prepare to guarantee that they have any combination ready that someone might ask for? You might ask a couple of questions. First, can you get double of one topping? That obviously changes the answer. Second, does the order of the toppings matter? ...
Arithmetic Sequences
... An arithmetic sequence is determined completely by the first term a, and the common difference d. Thus, if we know the first two terms of an arithmetic sequence, then we can find the equation for the nth term. Finding the Terms of an Arithmetic Sequence: Example 2: Find the nth term, the fifth term, ...
... An arithmetic sequence is determined completely by the first term a, and the common difference d. Thus, if we know the first two terms of an arithmetic sequence, then we can find the equation for the nth term. Finding the Terms of an Arithmetic Sequence: Example 2: Find the nth term, the fifth term, ...
1. Problems and Results in Number Theory
... X and I asked : Are there infinitely many 2k-tuples (k > 1) of consecutive primes pn +i < • • • < pn+2k satisfying pn+i + t = pn+k+i' for some t = t(k) and i = 1, . . . , k? The prime k-tuple conjecture of course implies this ; the point is to try to prove this without any hypotheses . We were unabl ...
... X and I asked : Are there infinitely many 2k-tuples (k > 1) of consecutive primes pn +i < • • • < pn+2k satisfying pn+i + t = pn+k+i' for some t = t(k) and i = 1, . . . , k? The prime k-tuple conjecture of course implies this ; the point is to try to prove this without any hypotheses . We were unabl ...
Situation 39: Summing Natural Numbers
... As before, the pairs are 1 + n, 2 + (n – 1), 3 + (n – 2), and so on. This time, there n !1 n +1 are pairs, each of which is n + 1, and one term, the middle term , is not ...
... As before, the pairs are 1 + n, 2 + (n – 1), 3 + (n – 2), and so on. This time, there n !1 n +1 are pairs, each of which is n + 1, and one term, the middle term , is not ...
Appendix A: Complex Numbers
... studied, where z is a complex variable. Many deep and beautiful theorems can be proved in this theory, one of which is the so-called fundamental theorem of algebra mentioned later (Theorem 5). We shall not pursue this here. The geometric description of the multiplication of two complex numbers follo ...
... studied, where z is a complex variable. Many deep and beautiful theorems can be proved in this theory, one of which is the so-called fundamental theorem of algebra mentioned later (Theorem 5). We shall not pursue this here. The geometric description of the multiplication of two complex numbers follo ...
Fibonacci notes
... Theorem 3.1 Let a1 and a2 be positive integers, and define a sequence (an ) by the Fibonacci recurrence: that is, an+2 = an + an+1 for n ≥ 1. Then there exist k, l, m such that am+n = Tk,l+n for all n ≥ 0. In other words, every sequence generated by the Fibonacci recurrence occurs, from some point o ...
... Theorem 3.1 Let a1 and a2 be positive integers, and define a sequence (an ) by the Fibonacci recurrence: that is, an+2 = an + an+1 for n ≥ 1. Then there exist k, l, m such that am+n = Tk,l+n for all n ≥ 0. In other words, every sequence generated by the Fibonacci recurrence occurs, from some point o ...
Real Numbers and Closure
... Like the counting numbers, the integers are closed under addition and multiplication. Similarly, when you subtract one integer from another, the answer is always an integer. That is, the integers are also closed under subtraction. Rational numbers The set of rational numbers includes all integers an ...
... Like the counting numbers, the integers are closed under addition and multiplication. Similarly, when you subtract one integer from another, the answer is always an integer. That is, the integers are also closed under subtraction. Rational numbers The set of rational numbers includes all integers an ...
1) - Mu Alpha Theta
... An ant decides to walk in a very interesting pattern. It first walks 1 meter north, stops, and then walks southeast until it is exactly 1 meter east of its starting location. It then walks 0.5 meters north, stops, and then walks southeast again until it is 0.5 meters east of the second starting loca ...
... An ant decides to walk in a very interesting pattern. It first walks 1 meter north, stops, and then walks southeast until it is exactly 1 meter east of its starting location. It then walks 0.5 meters north, stops, and then walks southeast again until it is 0.5 meters east of the second starting loca ...
