9.6 Learning to Count 101

9.6 Learning How to Count in Pre-Calculus

First off know your number line.

For a reference:

-5,-4,-3,-1, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 99, 100, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 120, 121, 122, 123, 124, 125, 126, 127, 128, 129, 130, 131, 132, 133, 134, 135, 136, 137, 138, 139, 140, 141, 142, 143, 144, 145, 146, 147, 148, 149, 150, 151, 152, 153, 154, 155, 156, 157, 158, 159, 160, 161, 162, 163, 164, 165, 166, 167, 168, 169, 170, 171, 172, 173, 174, 175, 176, 177, 178, 179, 180, 181, 182, 183, 184, 185, 186, 187, 188, 189, 190, 191, 192, 193, 194, 195, 196, 197, 198, 199, 200, 201, 202, 203, 204, 205, 206, 207, 208, 209, 210, 211, 212, 213, 214, 215, 216, 217, 218, 219, 220, 221, 222, 223, 224, 225, 226, 227, 228, 229, 230, 231, 232, 233, 234, 235, 236, 237, 238, 239, 240, 241, 242, 243, 244, 245, 246, 247, 248, 249, 250, 251, 252, 253, 254, 255, 256, 257, 258, 259, 260, 261, 262, 263, 264, 265, 266, 267, 268, 269, 270, 271, 272, 273, 274, 275, 276, 277, 278, 279, 280, 281, 282, 283, 284, 285, 286, 287, 288, 289, 290, 291, 292, 293, 294, 295, 296, 297, 298, 299, 300, 301, 302, 303, 304, 305, 306, 307, 308, 309, 310, 311, 312, 313, 314, 315, 316, 317, 318, 319, 320, 321, 322, 323, 324, 325, 326, 327, 328, 329, 330, 331, 332, 333, 334, 335, 336, 337, 338, 339, 340, 341, 342, 343, 344, 345, 346, 347, 348, 349, 350, 351, 352, 353, 354, 355, 356, 357, 358, 359, 360, 361, 362, 363, 364, 365, 366, 367, 368, 369, 370, 371, 372, 373, 374, 375, 376, 377, 378, 379, 380, 381, 382, 383, 384, 385, 386, 387, 388, 389, 390, 391, 392, 393, 394, 395, 396, 397, 398, 399, 400, 401, 402, 403, 404, 405, 406, 407, 408, 409, 410, 411, 412, 413, 414, 415, 416, 417, 418, 419, 420, 421, 422, 423, 424, 425, 426, 427, 428, 429, 430, 431, 432, 433, 434, 435, 436, 437, 438, 439, 440, 441, 442, 443, 444, 445, 446, 447, 448, 449, 450, 451, 452, 453, 454, 455, 456, 457, 458, 459, 460, 461, 462, 463, 464, 465, 466, 467, 468, 469, 470, 471, 472, 473, 474, 475, 476, 477, 478, 479, 480, 481, 482, 483, 484, 485, 486, 487, 488, 489, 490, 491, 492, 493, 494, 495, 496, 497, 498, 499, 500, 501, 502, 503, 504, 505, 506, 507, 508, 509, 510, 511, 512, 513, 514, 515, 516, 517, 518, 519, 520, 521, 522, 523, 524, 525, 526, 527, 528, 529, 530, 531, 532, 533, 534, 535, 536, 537, 538, 539, 540, 541, 542, 543, 544, 545, 546, 547, 548, 549, 550, 551, 552, 553, 554, 555, 556, 557, 558, 559, 560, 561, 562, 563, 564, 565, 566, 567, 568, 569, 570, 571, 572, 573, 574, 575, 576, 577, 578, 579, 580, 581, 582, 583, 584, 585, 586, 587, 588, 589, 590, 591, 592, 593, 594, 595, 596, 597, 598, 599, 600, 601, 602, 603, 604, 605, 606, 607, 608, 609, 610, 611, 612, 613, 614, 615, 616, 617, 618, 619, 620, 621, 622, 623, 624, 625, 626, 627, 628, 629, 630, 631, 632, 633, 634, 635, 636, 637, 638, 639, 640, 641, 642, 643, 644, 645, 646, 647, 648, 649, 650, 651, 652, 653, 654, 655, 656, 657, 658, 659, 660, 661, 662, 663, 664, 665, 666, 667, 668, 669, 670, 671, 672, 673, 674, 675, 676, 677, 678, 679, 680, 681, 682, 683, 684, 685, 686, 687, 688, 689, 690, 691, 692, 693, 694, 695, 696, 697, 698, 699, 700, 701, 702, 703, 704, 705, 706, 707, 708, 709, 710, 711, 712, 713, 714, 715, 716, 717, 718, 719, 720, 721, 722, 723, 724, 725, 726, 727, 728, 729, 730, 731, 732, 733, 734, 735, 736, 737, 738, 739, 740, 741, 742, 743, 744, 745, 746, 747, 748, 749, 750, 751, 752, 753, 754, 755, 756, 757, 758, 759, 760, 761, 762, 763, 764, 765, 766, 767, 768, 769, 770, 771, 772, 773, 774, 775, 776, 777, 778, 779, 780, 781, 782, 783, 784, 785, 786, 787, 788, 789, 790, 791, 792, 793, 794, 795, 796, 797, 798, 799, 800, 801, 802, 803, 804, 805, 806, 807, 808, 809, 810, 811, 812, 813, 814, 815, 816, 817, 818, 819, 820, 821, 822, 823, 824, 825, 826, 827, 828, 829, 830, 831, 832, 833, 834, 835, 836, 837, 838, 839, 840, 841, 842, 843, 844, 845, 846, 847, 848, 849, 850, 851, 852, 853, 854, 855, 856, 857, 858, 859, 860, 861, 862, 863, 864, 865, 866, 867, 868, 869, 870, 871, 872, 873, 874, 875, 876, 877, 878, 879, 880, 881, 882, 883, 884, 885, 886, 887, 888, 889, 890, 891, 892, 893, 894, 895, 896, 897, 898, 899, 900, 901, 902, 903, 904, 905, 906, 907, 908, 909, 910, 911, 912, 913, 914, 915, 916, 917, 918, 919, 920, 921, 922, 923, 924, 925, 926, 927, 928, 929, 930, 931, 932, 933, 934, 935, 936, 937, 938, 939, 940, 941, 942, 943, 944, 945, 946, 947, 948, 949, 950, 951, 952, 953, 954, 955, 956, 957, 958, 959, 960, 961, 962, 963, 964, 965, 966, 967, 968, 969, 970, 971, 972, 973, 974, 975, 976, 977, 978, 979, 980, 981, 982, 983, 984, 985, 986, 987, 988, 989, 990, 991, 992, 993, 994, 995, 996, 997, 998, 999, 1000,

To start off with some examples,

Ex. 1. If you have 4 different types of cheese to put on a pizza, and 4 different types of toppings, how many combinations can you create.

1 and 1    2 and 1    3 and 1   4 and 1

1 and 2    2 and 2    3 and 2   4 and 2
1 and 3    2 and 3    3 and 3   4 and 3
1 and 4    2 and 4    3 and 4   4 and 4

That creates 16 combinations. It could simply be solved by 4 · 4 = 16.


Ex. 2. If can wear 10 different wigs or 15 different natural hairstyles, how many hair styles can you choose from?

Well you could wear:
Wig #1    Wig #6    Hairstyle #1    Hairstyle #6    Hairstyle #11
Wig #2    Wig #7    Hairstyle #2    Hairstyle #7    Hairstyle #12
Wig #3    Wig #8    Hairstyle #3    Hairstyle #8    Hairstyle #13
Wig #4    Wig #9    Hairstyle #4    Hairstyle #9    Hairstyle #14
Wig #5    Wig #10  Hairstyle #5    Hairstyle #10   Hairstyle #15

And that is 25 different hairstyles which could be solved by 10 + 15 = 25.

If you would like, multiplication can be determined by "and" and addition can be determined with "or".

Permutations

A permutation contains different numbers which are assigned a position such as first, second or third.

This is when factorials come into place.

Ex 1. In the WCSR (World Championship of Snail Racers), there are 7 snails racing. What are the possible outcomes of the race.

So any of the 7 could win first, so 7.

Any of the remaining could win second, so 6.

Then the 5 remaining for the rest of the positions, so 5.

And so on for 4, 3, 2, 1.

You could then determine it is 7 · 6 · 5 · 4 · 3 · 2 · 1.

Which miraculously is 7!! (exclamation point at the end of a factorial, bam)

=5040

So there are 5040 possible outcomes of the race. Not including ties of course.

Ex 2. Now if the problem is getting tricky, such as it asks you for the possible outcomes for first, second and third.

But when they are trying to trick you, it simply means mind tricks, and you can out trick the mind tricks by tricking your mind into tricking the mind tricks into non-mind tricks, which is what the problem is asking you to do.

By following that, you can determine the same principle above, you can infer,

Any of the 7 could win first, so 7

Any of the 6 remaining could win second, so 6

And any of the 5 remaining could win third, so 5

That means simply 7 · 6 · 5

Which equals 210 possible combinations.

Which introduces the principle of    news.

Where n is the number of snails, and r  is the number of positions you are looking for.

Combinations

Combinations are similar to permutations in that they calculate the possible number of times something could occur.

The difference is that in combinations order does not matter.

In my opinion you could take this two different ways

1.  Order does not matter, therefore ABC is different than CBA, because they are in different order and it doesn't matter, they both count.

2. Order does not matter, therefore even if ABC and CBA are in different orders, it doesn't matter, they don't count.

But apparently it is number 2.

So even if they are in different orders, the same combination of letters would not count.

Ex. 1. You have to type a paper only using the letters R S T L N E (anyone?), how many combinations of words could you make using each letter once for a 6 letter word (order does not matter)?

So we have 6 letters, how many word combinations can we make with 4 letters.

We get 15 possible word combinations.
Wrong, its 4.
Theres only 4 words, the rest aren't words. :/

9.4 Mathematical Induction

Mathematical Induction:
Induction can be used to prove statements involving a positive integer n.

A proof must be used to verify that a rule, pattern, or formula is valid for all values of n.

This is called the Principle of Mathematical Induction:

Let  be a statement involving the positive integer n.

In order to show a statement is true for all values of n, you must follow these steps:

 

1. Show that the statement is true for n=1.

2. Assume that the statement is true for some integer n.

3. Use the above assumption to prove that the statement is valid for the next integer n+1.

 

 

Example: Prove the following formula using mathematical induction:

 





Using the steps of mathematical induction, we must:

1. Show it's true for n=1.

 

 

 

2. Assume that the statement is true for some integer n.

 

 

 

3. Prove the statement is valid for n+1.

This means proving that this formula is true: 

 

 



The next step is to replace the first part of the equation with

The result is:

This equation can be factored to equal this, which was our goal for this step:

 We have now proved that the formula:

 

is true when n=1, assumed that it is true for all values of n, and proven that it is true for the next integer n+1.

 

 

 

Sums of Powers of Integers

These formulas can be used to find the sums of various powers of the first n positive integers.

 

 

 

 

 

 

 

 

 
 
 

Finite Differences

There are two types of finite differences:

First differences are the differences between consecutive terms of a sequence.

Second differences are the differences between the first differences.

 

Example: 1, 4, 9, 16, 25

First: 3, 5, 7, 9

Second: 2, 2, 2

 

If the second differences are the same and nonzero, the sequence is a quadratic model. If the first differences are the same and nonzero, it is a linear model. If none of the differences are the same, it is neither model.

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 



9.5 The Binomial Theorem

A binomial is a polynomial that has two terms.  The binomial theorem gives you a mechanical way to expand a binomial raised to a certain power.

You can expandto several powers of n, and make several observations for patterns to help you remember the theorem and understand it better.  Let’s look at this binomial raised to several different powers:

• ·      In each binomial expansion, there are (n+1) terms.  So if you are raising a binomial to the 7^th power, there will be 8 terms when fully expanded.
• ·      In each expansion, the x power begins with the power the binomial is raised to and descends in each term until it is raised to the 0^th power.  The y power begins with 0 and ascends each term until it is raised to the power the whole binomial is being raised to.
• ·      The sum of the powers in each term is n.  So no matter what term you are looking at in the expanded form, the exponents of x and y added together will be the number the whole binomial is being raised to.
• ·      The coefficients of each term increase and decrease symmetrically.  If you look at the 5^th row, the coefficients are “1, 5, 10, 10, 5, 1”.  Whether you start at the beginning or end, the coefficients will be in the same order and be symmetrical.

The Binomial Theorem

-In the expansion of the coefficient of is

 **** The symbol     is often used in place of , to denote binomial coefficients.

Pascal’s Triangle and Its Patterns

The first and last number in each row is 1.  Every other number in each row is found by adding the two numbers immediately above the number.  The numbers in this triangle are exactly the same numbers as the coefficients of binomial expansions.

The first diagonal are all 1’s.  The second are counting numbers.  The next two diagonals are triangular numbers and tetrahedral numbers, respectively.

All of the horizontal sums are powers of 2!

Each line is 11 raised to some power.  When you get to the 5th row, the digits just overlap!

Pictures of Pascal's Triangle courtesy of mathisfun.com: http://www.mathsisfun.com/pascals-triangle.html 

Expanding a Binomial

 Write the expansion for : Use the third row of Pascal's Triangle:

9.3 Geometric Sequences and Series!

CLARIFICATION

Sequence- commas

2, 4, 8, 16...

Series- plus signs/ addition

2 + 4 + 8 + 16...

Geometric sequence- ratios of consecutive terms are the same

Example:       5, 15, 45, 135, 405...           r = 3

The number r is the common ratio.  It cannot be zero, otherwise, it is not a geometric sequence. When r is negative, the signs will alternate.  This means that if the first term is positive, the next will be negative, then positive, then negative and so on.  

(The common difference of an arithmetic sequence is like the common ratio of a geometric sequence)

Recursive Formula

Explicit Formula (exponential)

Example: find the nth term 

Sum of Finite Geometric Sequences

Sum of an Infinite Geometric Series

We can only find the sum of an infinite geometric series if .  If the common ratio was greater than, then the sum would be infinity.  This is because adding an infinite amount of continuously increasing numbers together will be infinity.  

This is the graph of a geometric sequence with a common ratio less than one (2/3).  The red line represents the numbers in the sequence.  The blue line represents the sum of the numbers.  See how the blue line begins to curve then levels off?  This is because the numbers keep getting smaller and smaller.