heureka

Please give the next 4 terms of this sequence. Thanks for any help:

2, 4, 7, 10, 15, 18, 23, 26............640 (is 100th term)

Please give the next 4 terms of this sequence. Thanks for any help:

2, 4, 7, 10, 15, 18, 23, 26............640 (is 100th term)

We need the prime number table

\(\text{The prime numbers. }\\ \begin{array}{|r|r||r|r|||r|r|||r|r|||r|r||r|r|} \hline n & p(n) & n & p(n) & n & p(n)& n & p(n)& n & p(n)& n & p(n) \\ \hline 1 & 2 &21 & 73 &41 & 179 & 61 & 283 &81 & 419 & 101 & 547 \\ 2 & 3 &22 & 79 &42 & 181 & 62 & 293 &82 & 421 & \cdots & \cdots \\ 3 & 5 &23 & 83 &43 & 191 & 63 & 307 &83 & 431 & \cdots & \cdots \\ 4 & 7 &24 & 89 &44 & 193 & 64 & 311 &84 & 433 & \cdots & \cdots \\ 5 & 11 &25 & 97 &45 & 197 & 65 & 313 &85 & 439 & \cdots & \cdots \\ 6 & 13 &26 & 101 &46 & 199 & 66 & 317 &86 & 443 & \cdots & \cdots \\ 7 & 17&27 & 103 &47 & 211 & 67 & 331 &87 & 449 & \cdots & \cdots \\ 8 & 19 &28 & 107 &48 & 223 & 68 & 337 &88 & 457 & \cdots & \cdots \\ 9 & 23 &29 & 109 &49 & 227 & 69 & 347 &89 & 461 & \cdots & \cdots \\ 10 & 29&30 & 113 &50 & 229 & 70 & 349 &90 & 463 & \cdots & \cdots \\ 11 & 31& 31 & 127& 51 & 233& 71 & 353 &91 & 467 & \cdots & \cdots \\ 12 & 37&32 & 131 &52 & 239 & 72 & 359 &92 & 479 & \cdots & \cdots \\ 13 & 41& 33 & 137 &53 & 241& 73 & 367 &93 & 487 & \cdots & \cdots \\ 14 & 43& 34 & 139 &54 & 251& 74 & 373 &94 & 491 & \cdots & \cdots \\ 15 & 47& 35 & 149 &55 & 257& 75 & 379 &95 & 499 & \cdots & \cdots \\ 16 & 53& 36 & 151 &56 & 263& 76 & 383 &96 & 503 & \cdots & \cdots \\ 17 & 59& 37 & 157 &57 & 269& 77 & 389 &97 & 509 & \cdots & \cdots \\ 18 & 61& 38 & 163 &58 & 271& 78 & 397 &98 & 521 & \cdots & \cdots \\ 19 & 67& 39 & 167 &59 & 277& 79 & 401 &99 & 523 & \cdots & \cdots \\ 20 & 71& 40 & 173 &60 & 281& 80 & 409 &100 & 541 & \cdots & \cdots \\ \hline \end{array} \)

The formula is \(\boxed{~ a_n = a_{n-1} + p(n) -p(n-1) +1 \qquad n \ge 2 \qquad a_1 = 2 ~}\)

\(\begin{array}{lcl} a_1 &=& 2 \\ \hline a_2 &=& a_{1} + p(2) -p(1) +1 \\ &=& 2 +3-2+1 \\ &=& 4 \\ \hline a_3 &=& a_{2} + p(3) -p(2) +1 \\ &=& 4 +5-3+1 \\ &=& 7 \\ \hline a_4 &=& a_{3} + p(4) -p(3) +1 \\ &=& 7 +7-5+1 \\ &=& 10 \\ \hline a_5 &=& a_{4} + p(5) -p(4) +1 \\ &=& 10 +11-7+1 \\ &=& 15 \\ \hline a_6 &=& a_{5} + p(6) -p(5) +1 \\ &=& 15 +13-11+1 \\ &=& 18 \\ \hline a_7 &=& a_{6} + p(7) -p(6) +1 \\ &=& 18 +17-13+1 \\ &=& 23 \\ \hline a_8 &=& a_{7} + p(8) -p(7) +1 \\ &=& 23 +19-17+1 \\ &=& 26 \\ \hline \hline a_9 &=& a_{8} + p(9) -p(8) +1 \\ &=& 26 +23-19+1 \\ &=& 31 \\ \hline a_{10} &=& a_{9} + p(10) -p(9) +1 \\ &=& 31 +29-23+1 \\ &=& 38 \\ \hline a_{11} &=& a_{10} + p(11) -p(10) +1 \\ &=& 38 +31-29+1 \\ &=& 41 \\ \hline a_{12} &=& a_{11} + p(12) -p(11) +1 \\ &=& 41 +37-31+1 \\ &=& 48 \\ \hline \cdots \\ \hline a_{100} &=& a_{99} + p(100) -p(99) +1 \\ &=& 621 +541-523+1 \\ &=& 640 \\ \hline \end{array} \)

\(a_1 \cdots a_{50}\\ \begin{array}{|r|r||r|r|||r|r|||r|r|||r|r|} \hline n & a(n) & n & a(n) & n & a(n)& n & a(n)& n & a(n)\\ \hline 1 & 2 & 11 & 41 & 21 & 93 &31 & 157 &41 & 219\\ 2 & 4 & 12 & 48 & 22 & 100 &32 & 162&42 & 222\\ 3 & 7 & 13 & 53 & 23 & 105 &33 & 169&43 & 233\\ 4 & 10 & 14 & 56 & 24 & 112 &34 & 172&44 & 236\\ 5 & 15 & 15 & 61 & 25 & 121 &35 & 183&45 & 241\\ 6 & 18 & 16 & 68 & 26 & 126 &36 & 186&46 & 244\\ 7 & 23 & 17 & 75 & 27 & 129 &37 & 193&47 & 257\\ 8 & 26 & 18 & 78 & 28 & 134 &38 & 200&48 & 270\\ 9 & 31 & 19 & 85 & 29 & 137 &39 & 205&49 & 275\\ 10 & 38 & 20 & 90 & 30 & 142 &40 & 212&50 & 278\\ \hline \end{array} \)

\(a_{51} \cdots a_{101}\\ \begin{array}{|r|r||r|r|||r|r|||r|r|||r|r||r|r|} \hline n & a(n) & n & a(n) & n & a(n)& n & a(n)& n & a(n)& n & a(n) \\ \hline 51 & 283 &61 & 343&71 & 423&81 & 499&91 & 557&101 & 647 \\ 52 & 290 &62 & 354&72 & 430&82 & 502&92 & 570&\cdots & \cdots \\ 53 & 293 &63 & 369&73 & 439&83 & 513&93 & 579&\cdots & \cdots \\ 54 & 304 &64 & 374&74 & 446&84 & 516&94 & 584&\cdots & \cdots \\ 55 & 311 &65 & 377&75 & 453&85 & 523&95 & 593&\cdots & \cdots \\ 56 & 318 &66 & 382&76 & 458&86 & 528&96 & 598&\cdots & \cdots \\ 57 & 325 &67 & 397&77 & 465&87 & 535&97 & 605&\cdots & \cdots \\ 58 & 328 &68 & 404&78 & 474&88 & 544&98 & 618&\cdots & \cdots \\ 59 & 335 &69 & 415&79 & 479&89 & 549&99 & 621&\cdots & \cdots \\ 60 & 340 &70 & 418&80 & 488&90 & 552&100 & 640&\cdots & \cdots \\ \hline \end{array} \)

binomialcdf( 7, 0.2, 1 ) = 0.5767168

\(\begin{array}{lcl} \boxed{~ \text{binomialcdf}(n,p,k) = \sum \limits_{i=0}^{k} \binom{n}{i} p^i(1-p)^{n-i} ~} \end{array} \\ \begin{array}{lcl} binomialcdf(7,0.2,1) &=& \sum \limits_{i=0}^{1} \binom{7}{i} 0.2^i(1-0.2)^{7-i} \\ &=& \sum \limits_{i=0}^{1} \binom{7}{i} 0.2^i\cdot 0.8^{7-i} \\ &=& \binom{7}{0} 0.2^0\cdot 0.8^{7-0}+\binom{7}{1} 0.2^1\cdot 0.8^{7-1} \\ &=& \binom{7}{0} 1\cdot 0.8^{7}+\binom{7}{1} 0.2^1\cdot 0.8^{6} \\ &=& \binom{7}{0} 0.8^{7}+\binom{7}{1} 0.2\cdot 0.8^{6} \qquad | \qquad \binom{7}{0} = 1 \qquad \binom{7}{1} = 7\\ &=& 1\cdot 0.8^{7}+7\cdot 0.2\cdot 0.8^{6}\\ &=& 0.8^{7}+1.4\cdot 0.8^{6}\\ &=& 0.2097152+ 0.3670016 \\ &=& 0.5767168 \\ &=& 57.67\ \% \end{array} \)

\(\begin{array}{lcl} p\approx 2\pi\sqrt{\frac12 (a^2+b^2)} \qquad a \text{ and } b \text{ are the half-axes !!!} \\ \text{ so } a = \frac{w}{2} \text{ and } b = \frac{h}{2} \\ \text{ we have } p\equiv 2\pi\sqrt{\frac12 [ (\frac{w}{2})^2 + (\frac{h}{2})^2 ] } \end{array} \\ \begin{array}{lcl} p &\approx& 2\pi\sqrt{\frac12 [ (\frac{w^2}{4}) + (\frac{h^2}{4}) ] }\\ p &\approx& 2\pi\sqrt{\frac12 (\frac{w^2+h^2}{4}) }\\ p &\approx& 2\pi\sqrt{\frac24 (\frac{w^2+h^2}{4}) }\\ p &\approx& 2\pi \cdot \frac{1}{4} \sqrt{2\cdot (w^2+h^2 ) }\\ p &\approx& \frac{\pi}{2}\cdot \sqrt{2\cdot (w^2+h^2 ) }\\ \end{array} \)

Mai is putting gold foil around the outside of an elliptical picture frame. The permieter of an ellipse is given by the formula p=3.14/2 Squareroot(2(h^2+w^2), where h is the height and w is the width, as shown in the diagram below. If an elliptical frame has an outside height equal to 4 inches nd an outside width equal to 3 inches. What is its outside perimeter, in inches?

So, What im getting is plug in the number for the variables?

\(\begin{array}{rcl} p &=& \frac{\pi}{2} \cdot \sqrt{ 2 \cdot \left(~ w^2+h^2 ~\right) } \qquad w = 3\ \text{inch} \qquad h = 4\ \text{inch} \\ p &=& \frac{\pi}{2} \cdot \sqrt{ 2 \cdot \left(~ 3^2+4^2 ~\right) } \\ p &=& \frac{\pi}{2} \cdot \sqrt{ 2 \cdot \left(~ 5^2 ~\right) } \qquad 3^2+4^2 = 5^2 \\ p &=& \frac{\pi}{2} \cdot 5\cdot \sqrt{ 2 } \\ p &=& \pi \cdot 2.5 \cdot \sqrt{ 2 } \\ \mathbf{p }& \mathbf{=} & \mathbf{11.1072073454}\ \text{inch} \end{array} \)

we have two milk pots 1st is 3kg and 2nd is 5kg so how can we measure exact 4kg without using any 3rd pot?

1. fill up the 3kg milk pot.

2. put it in the 5kg milk pot. The 5 kg pot now contains 3 kg

3. fill up the 3kg milk pot again.

4. put it in the 5kg milk pot. The 5 kg pot now contains 5 kg. The 3kg milk pot now contains 1kg.

5. void the 5kg milk pot.

6. put the 1kg (of the 3kg milk pot) in the 5kg milk pot. The 5 kg pot now contains 1 kg.

7. fill up the 3kg milk pot.

8 put it in the 5 kg mild pot. The 5 kg pot now contains 4 kg.

Using US coins of 1¢, 5¢, 10¢, 25¢, 50¢, and $1, how many ways are there to make change for $1.00.  List all the ways.

\(\small{ \begin{array}{|r|r|r|r|r|r|r|r|} \hline Number & & 1~ \text{Cent} & 5~ \text{Cent} & 10~ \text{Cent} & 25~ \text{Cent} & 50~ \text{Cent} & $1\\ \hline 1. & 100 = &&&&&& 1\\ 2. & 100 = &&&&& 2\\ 3. & 100 = &&&& 2& 1\\ 4. & 100 = &&&& 4&\\ 5. & 100 = &&& 5&& 1\\ 6. & 100 = &&& 5& 2&\\ 7. & 100 = &&& 10&&\\ 8. & 100 = && 1& 2& 1& 1\\ 9. & 100 = && 1& 2& 3&\\ 10. & 100 = && 1& 7& 1&\\ 11. & 100 = && 2& 4&& 1\\ 12. & 100 = && 2& 4& 2&\\ 13. & 100 = && 2& 9&&\\ 14. & 100 = && 3& 1& 1& 1\\ 15. & 100 = && 3& 1& 3&\\ 16. & 100 = && 3& 6& 1&\\ 17. & 100 = && 4& 3&& 1\\ 18. & 100 = && 4& 3& 2&\\ 19. & 100 = && 4& 8&&\\ 20. & 100 = && 5&& 1& 1\\ 21. & 100 = && 5&& 3&\\ 22. & 100 = && 5& 5& 1&\\ 23. & 100 = && 6& 2&& 1\\ 24. & 100 = && 6& 2& 2&\\ 25. & 100 = && 6& 7&&\\ 26. & 100 = && 7& 4& 1&\\ 27. & 100 = && 8& 1&& 1\\ 28. & 100 = && 8& 1& 2&\\ 29. & 100 = && 8& 6&&\\ 30. & 100 = && 9& 3& 1&\\ 31. & 100 = && 10&&& 1\\ 32. & 100 = && 10&& 2&\\ 33. & 100 = && 10& 5&&\\ 34. & 100 = && 11& 2& 1&\\ 35. & 100 = && 12& 4&&\\ 36. & 100 = && 13& 1& 1&\\ 37. & 100 = && 14& 3&&\\ 38. & 100 = && 15&& 1&\\ 39. & 100 = && 16& 2&&\\ 40. & 100 = && 18& 1&&\\ 41. & 100 = && 20&&&\\ 42. & 100 = & 5&& 2& 1& 1\\ 43. & 100 = & 5&& 2& 3&\\ 44. & 100 = & 5&& 7& 1&\\ 45. & 100 = & 5& 1& 4&& 1\\ 46. & 100 = & 5& 1& 4& 2&\\ 47. & 100 = & 5& 1& 9&&\\ 48. & 100 = & 5& 2& 1& 1& 1\\ 49. & 100 = & 5& 2& 1& 3&\\ 50. & 100 = & 5& 2& 6& 1&\\ 51. & 100 = & 5& 3& 3&& 1\\ 52. & 100 = & 5& 3& 3& 2&\\ 53. & 100 = & 5& 3& 8&&\\ 54. & 100 = & 5& 4&& 1& 1\\ 55. & 100 = & 5& 4&& 3&\\ 56. & 100 = & 5& 4& 5& 1&\\ 57. & 100 = & 5& 5& 2&& 1\\ 58. & 100 = & 5& 5& 2& 2&\\ 59. & 100 = & 5& 5& 7&&\\ 60. & 100 = & 5& 6& 4& 1&\\ 61. & 100 = & 5& 7& 1&& 1\\ 62. & 100 = & 5& 7& 1& 2&\\ 63. & 100 = & 5& 7& 6&&\\ 64. & 100 = & 5& 8& 3& 1&\\ 65. & 100 = & 5& 9&&& 1\\ 66. & 100 = & 5& 9&& 2&\\ 67. & 100 = & 5& 9& 5&&\\ 68. & 100 = & 5& 10& 2& 1&\\ 69. & 100 = & 5& 11& 4&&\\ 70. & 100 = & 5& 12& 1& 1&\\ 71. & 100 = & 5& 13& 3&&\\ 72. & 100 = & 5& 14&& 1&\\ 73. & 100 = & 5& 15& 2&&\\ 74. & 100 = & 5& 17& 1&&\\ 75. & 100 = & 5& 19&&&\\ 76. & 100 = & 10&& 4&& 1\\ 77. & 100 = & 10&& 4& 2&\\ 78. & 100 = & 10&& 9&&\\ 79. & 100 = & 10& 1& 1& 1& 1\\ 80. & 100 = & 10& 1& 1& 3&\\ 81. & 100 = & 10& 1& 6& 1&\\ 82. & 100 = & 10& 2& 3&& 1\\ 83. & 100 = & 10& 2& 3& 2&\\ 84. & 100 = & 10& 2& 8&&\\ 85. & 100 = & 10& 3&& 1& 1\\ 86. & 100 = & 10& 3&& 3&\\ 87. & 100 = & 10& 3& 5& 1&\\ 88. & 100 = & 10& 4& 2&& 1\\ 89. & 100 = & 10& 4& 2& 2&\\ 90. & 100 = & 10& 4& 7&&\\ 91. & 100 = & 10& 5& 4& 1&\\ 92. & 100 = & 10& 6& 1&& 1\\ 93. & 100 = & 10& 6& 1& 2&\\ 94. & 100 = & 10& 6& 6&&\\ 95. & 100 = & 10& 7& 3& 1&\\ 96. & 100 = & 10& 8&&& 1\\ 97. & 100 = & 10& 8&& 2&\\ 98. & 100 = & 10& 8& 5&&\\ 99. & 100 = & 10& 9& 2& 1&\\ 100. & 100 = & 10& 10& 4&&\\ 101. & 100 = & 10& 11& 1& 1&\\ 102. & 100 = & 10& 12& 3&&\\ 103. & 100 = & 10& 13&& 1&\\ 104. & 100 = & 10& 14& 2&&\\ 105. & 100 = & 10& 16& 1&&\\ 106. & 100 = & 10& 18&&&\\ 107. & 100 = & 15&& 1& 1& 1\\ 108. & 100 = & 15&& 1& 3&\\ 109. & 100 = & 15&& 6& 1&\\ 110. & 100 = & 15& 1& 3&& 1\\ 111. & 100 = & 15& 1& 3& 2&\\ 112. & 100 = & 15& 1& 8&&\\ 113. & 100 = & 15& 2&& 1& 1\\ 114. & 100 = & 15& 2&& 3&\\ 115. & 100 = & 15& 2& 5& 1&\\ 116. & 100 = & 15& 3& 2&& 1\\ 117. & 100 = & 15& 3& 2& 2&\\ 118. & 100 = & 15& 3& 7&&\\ 119. & 100 = & 15& 4& 4& 1&\\ 120. & 100 = & 15& 5& 1&& 1\\ 121. & 100 = & 15& 5& 1& 2&\\ 122. & 100 = & 15& 5& 6&&\\ 123. & 100 = & 15& 6& 3& 1&\\ 124. & 100 = & 15& 7&&& 1\\ 125. & 100 = & 15& 7&& 2&\\ 126. & 100 = & 15& 7& 5&&\\ 127. & 100 = & 15& 8& 2& 1&\\ 128. & 100 = & 15& 9& 4&&\\ 129. & 100 = & 15& 10& 1& 1&\\ 130. & 100 = & 15& 11& 3&&\\ 131. & 100 = & 15& 12&& 1&\\ 132. & 100 = & 15& 13& 2&&\\ 133. & 100 = & 15& 15& 1&&\\ 134. & 100 = & 15& 17&&&\\ 135. & 100 = & 20&& 3&& 1\\ 136. & 100 = & 20&& 3& 2&\\ 137. & 100 = & 20&& 8&&\\ 138. & 100 = & 20& 1&& 1& 1\\ 139. & 100 = & 20& 1&& 3&\\ 140. & 100 = & 20& 1& 5& 1&\\ 141. & 100 = & 20& 2& 2&& 1\\ 142. & 100 = & 20& 2& 2& 2&\\ 143. & 100 = & 20& 2& 7&&\\ 144. & 100 = & 20& 3& 4& 1&\\ 145. & 100 = & 20& 4& 1&& 1\\ 146. & 100 = & 20& 4& 1& 2&\\ 147. & 100 = & 20& 4& 6&&\\ 148. & 100 = & 20& 5& 3& 1&\\ 149. & 100 = & 20& 6&&& 1\\ 150. & 100 = & 20& 6&& 2&\\ 151. & 100 = & 20& 6& 5&&\\ 152. & 100 = & 20& 7& 2& 1&\\ 153. & 100 = & 20& 8& 4&&\\ 154. & 100 = & 20& 9& 1& 1&\\ 155. & 100 = & 20& 10& 3&&\\ 156. & 100 = & 20& 11&& 1&\\ 157. & 100 = & 20& 12& 2&&\\ 158. & 100 = & 20& 14& 1&&\\ 159. & 100 = & 20& 16&&&\\ 160. & 100 = & 25&&& 1& 1\\ 161. & 100 = & 25&&& 3&\\ 162. & 100 = & 25&& 5& 1&\\ 163. & 100 = & 25& 1& 2&& 1\\ 164. & 100 = & 25& 1& 2& 2&\\ 165. & 100 = & 25& 1& 7&&\\ 166. & 100 = & 25& 2& 4& 1&\\ 167. & 100 = & 25& 3& 1&& 1\\ 168. & 100 = & 25& 3& 1& 2&\\ 169. & 100 = & 25& 3& 6&&\\ 170. & 100 = & 25& 4& 3& 1&\\ 171. & 100 = & 25& 5&&& 1\\ 172. & 100 = & 25& 5&& 2&\\ 173. & 100 = & 25& 5& 5&&\\ 174. & 100 = & 25& 6& 2& 1&\\ 175. & 100 = & 25& 7& 4&&\\ 176. & 100 = & 25& 8& 1& 1&\\ 177. & 100 = & 25& 9& 3&&\\ 178. & 100 = & 25& 10&& 1&\\ 179. & 100 = & 25& 11& 2&&\\ 180. & 100 = & 25& 13& 1&&\\ 181. & 100 = & 25& 15&&&\\ 182. & 100 = & 30&& 2&& 1\\ 183. & 100 = & 30&& 2& 2&\\ 184. & 100 = & 30&& 7&&\\ 185. & 100 = & 30& 1& 4& 1&\\ 186. & 100 = & 30& 2& 1&& 1\\ 187. & 100 = & 30& 2& 1& 2&\\ 188. & 100 = & 30& 2& 6&&\\ 189. & 100 = & 30& 3& 3& 1&\\ 190. & 100 = & 30& 4&&& 1\\ 191. & 100 = & 30& 4&& 2&\\ 192. & 100 = & 30& 4& 5&&\\ 193. & 100 = & 30& 5& 2& 1&\\ 194. & 100 = & 30& 6& 4&&\\ 195. & 100 = & 30& 7& 1& 1&\\ 196. & 100 = & 30& 8& 3&&\\ 197. & 100 = & 30& 9&& 1&\\ 198. & 100 = & 30& 10& 2&&\\ 199. & 100 = & 30& 12& 1&&\\ 200. & 100 = & 30& 14&&&\\ 201. & 100 = & 35&& 4& 1&\\ 202. & 100 = & 35& 1& 1&& 1\\ 203. & 100 = & 35& 1& 1& 2&\\ 204. & 100 = & 35& 1& 6&&\\ 205. & 100 = & 35& 2& 3& 1&\\ 206. & 100 = & 35& 3&&& 1\\ 207. & 100 = & 35& 3&& 2&\\ 208. & 100 = & 35& 3& 5&&\\ 209. & 100 = & 35& 4& 2& 1&\\ 210. & 100 = & 35& 5& 4&&\\ 211. & 100 = & 35& 6& 1& 1&\\ 212. & 100 = & 35& 7& 3&&\\ 213. & 100 = & 35& 8&& 1&\\ 214. & 100 = & 35& 9& 2&&\\ 215. & 100 = & 35& 11& 1&&\\ 216. & 100 = & 35& 13&&&\\ 217. & 100 = & 40&& 1&& 1\\ 218. & 100 = & 40&& 1& 2&\\ 219. & 100 = & 40&& 6&&\\ 220. & 100 = & 40& 1& 3& 1&\\ 221. & 100 = & 40& 2&&& 1\\ 222. & 100 = & 40& 2&& 2&\\ 223. & 100 = & 40& 2& 5&&\\ 224. & 100 = & 40& 3& 2& 1&\\ 225. & 100 = & 40& 4& 4&&\\ 226. & 100 = & 40& 5& 1& 1&\\ 227. & 100 = & 40& 6& 3&&\\ 228. & 100 = & 40& 7&& 1&\\ 229. & 100 = & 40& 8& 2&&\\ 230. & 100 = & 40& 10& 1&&\\ 231. & 100 = & 40& 12&&&\\ 232. & 100 = & 45&& 3& 1&\\ 233. & 100 = & 45& 1&&& 1\\ 234. & 100 = & 45& 1&& 2&\\ 235. & 100 = & 45& 1& 5&&\\ 236. & 100 = & 45& 2& 2& 1&\\ 237. & 100 = & 45& 3& 4&&\\ 238. & 100 = & 45& 4& 1& 1&\\ 239. & 100 = & 45& 5& 3&&\\ 240. & 100 = & 45& 6&& 1&\\ 241. & 100 = & 45& 7& 2&&\\ 242. & 100 = & 45& 9& 1&&\\ 243. & 100 = & 45& 11&&&\\ 244. & 100 = & 50&&&& 1\\ 245. & 100 = & 50&&& 2&\\ 246. & 100 = & 50&& 5&&\\ 247. & 100 = & 50& 1& 2& 1&\\ 248. & 100 = & 50& 2& 4&&\\ 249. & 100 = & 50& 3& 1& 1&\\ 250. & 100 = & 50& 4& 3&&\\ 251. & 100 = & 50& 5&& 1&\\ 252. & 100 = & 50& 6& 2&&\\ 253. & 100 = & 50& 8& 1&&\\ 254. & 100 = & 50& 10&&&\\ 255. & 100 = & 55&& 2& 1&\\ 256. & 100 = & 55& 1& 4&&\\ 257. & 100 = & 55& 2& 1& 1&\\ 258. & 100 = & 55& 3& 3&&\\ 259. & 100 = & 55& 4&& 1&\\ 260. & 100 = & 55& 5& 2&&\\ 261. & 100 = & 55& 7& 1&&\\ 262. & 100 = & 55& 9&&&\\ 263. & 100 = & 60&& 4&&\\ 264. & 100 = & 60& 1& 1& 1&\\ 265. & 100 = & 60& 2& 3&&\\ 266. & 100 = & 60& 3&& 1&\\ 267. & 100 = & 60& 4& 2&&\\ 268. & 100 = & 60& 6& 1&&\\ 269. & 100 = & 60& 8&&&\\ 270. & 100 = & 65&& 1& 1&\\ 271. & 100 = & 65& 1& 3&&\\ 272. & 100 = & 65& 2&& 1&\\ 273. & 100 = & 65& 3& 2&&\\ 274. & 100 = & 65& 5& 1&&\\ 275. & 100 = & 65& 7&&&\\ 276. & 100 = & 70&& 3&&\\ 277. & 100 = & 70& 1&& 1&\\ 278. & 100 = & 70& 2& 2&&\\ 279. & 100 = & 70& 4& 1&&\\ 280. & 100 = & 70& 6&&&\\ 281. & 100 = & 75&&& 1&\\ 282. & 100 = & 75& 1& 2&&\\ 283. & 100 = & 75& 3& 1&&\\ 284. & 100 = & 75& 5&&&\\ 285. & 100 = & 80&& 2&&\\ 286. & 100 = & 80& 2& 1&&\\ 287. & 100 = & 80& 4&&&\\ 288. & 100 = & 85& 1& 1&&\\ 289. & 100 = & 85& 3&&&\\ 290. & 100 = & 90&& 1&&\\ 291. & 100 = & 90& 2&&&\\ 292. & 100 = & 95& 1&&&\\ 293. & 100 = &100&&&&\\ \hline \end{array} } \)

What is this equal to? 1+4+9+16+25...

Don't say that it is infinity.

\(\begin{array}{lrrrrrrrrrr} & {\color{red}d_0 = 1^2} && 2^2 && 3^2 && 4^2 && 5^2 && \cdots \\ \text{1. Difference } && {\color{red}d_1 = 3} && 5 && 7 && 9 && 11 && \cdots \\ \text{2. Difference } &&& {\color{red}d_2 = 2} && 2 && 2 && 2 && 2 && \cdots \\ \end{array}\)

\(\begin{array}{rcl} s_n &=& \binom{n}{1}\cdot {\color{red}d_0 } + \binom{n}{2}\cdot {\color{red}d_1 } + \binom{n}{3}\cdot {\color{red}d_2 }\\ s_n &=& \binom{n}{1}\cdot {\color{red}1 } + \binom{n}{2}\cdot {\color{red}3} + \binom{n}{3}\cdot {\color{red}2}\\ \\ \hline \binom{n}{1} &=& n \\ \binom{n}{2} &=& ( \frac{n}{2} ) \cdot ( \frac{n-1}{1} ) \\ \binom{n}{3} &=& ( \frac{n}{3} ) \cdot ( \frac{n-1}{2} )\cdot ( \frac{n-2}{1} ) \\ \hline \\ s_n &=& (n)\cdot {\color{red}1} + ( \frac{n}{2} ) \cdot ( \frac{n-1}{1} )\cdot {\color{red}3} + ( \frac{n}{3} ) \cdot ( \frac{n-1}{2} )\cdot ( \frac{n-2}{1} )\cdot {\color{red}2} \quad | \quad \cdot 6\\ 6\cdot s_n &=& n\cdot 6 + n \cdot ( n-1 )\cdot 9 + n \cdot ( n-1 )\cdot ( n-2 )\cdot 2 \\ 6\cdot s_n &=& n \left[~ 6 + ( n-1 )\cdot 9 + ( n-1 )\cdot ( n-2 )\cdot 2 ~\right] \\ 6\cdot s_n &=& (n) \left[~ 6 + 9n-9 + (n^2 - 3n + 2)\cdot 2 ~\right] \\ 6\cdot s_n &=& (n) \left[~ -3 + 9n + (n^2 - 3n + 2)\cdot 2 ~\right] \\ 6\cdot s_n &=& (n) \left(~ -3 + 9n + 2n^2 - 6n + 4 ~\right) \\ 6\cdot s_n &=& (n) \left(~ 1 + 3n + 2n^2 ~\right) \\ 6\cdot s_n &=& (n) \cdot (n+1) \cdot ( 2n + 1 ) \\\\ \mathbf{s_n} &=& \mathbf{ \frac{ (n) \cdot (n+1) \cdot ( 2n + 1 ) }{6} } \\\\ s_1 &=& 1 = \frac{ 1 \cdot ( 1+1) \cdot ( 2\cdot 1 + 1 ) }{6} = 1\\ s_2 &=& 1+4 = \frac{ 2 \cdot ( 2+1) \cdot ( 2\cdot 2 + 1 ) }{6} = 5\\ s_3 &=& 1+4+9 = \frac{ 3 \cdot ( 3+1) \cdot ( 2\cdot 3 + 1 ) }{6} = 14\\ s_4 &=& 1+4+9+16 = \frac{ 4 \cdot ( 4+1) \cdot ( 2\cdot 4 + 1 ) }{6} = 30\\ s_5 &=& 1+4+9+16+25 = \frac{ 5 \cdot ( 5+1) \cdot ( 2\cdot 5 + 1 ) }{6} = 55\\ \cdots \end{array}\)

joey earns $7.65 an hour at his part time job.he received a 20% raise.how much will joey now earn for working 40 hours

\(\begin{array}{rcl} \frac{ $7.65 }{ \text{hour} } \cdot ( 100\%+20 \% ) \cdot 40\ \text{hours} &=& $7.65 \cdot 1.2 \cdot 40 \\ &=& $367.2 \end{array}\)

Joey will now earn for working 40 hours $367.2

Wie behandelt man den Winkel in einer Wurzel Beispiel : sqrt(40°)/(1/min x 0,573°)

Bei einem Winkel außerhalb einer Winkelfunktion verwendet man das Bogenmaß.

Beispiel:

\(E\quad =\quad M\;+\;e\cdot \sin E\)

Diese Gleichung nennt man die Keplergleichung.

Achtung: in der Herleitung wird vorausgesetzt, dass die Winkel im Bogenmaß gegeben sind.