+0  
 
0
313
5
avatar

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)

Guest Jan 14, 2016

Best Answer 

 #5
avatar+18712 
+15

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)

 

Thanks heureka: brilliant work!. In my book it is stated slightly different: It is the sequence of natual or counting numbers added to prime numbers - 1. So, we have:

Counting numbers: 1, 2, 3, 4, 5, 6............

Prime numbers    + 2, 3, 5, 7, 11, 13...............

Subtract 1  from      2, 4, 7, 10, 15, 18..........etc.

each term

 

\(\text{The formula is}\\ \boxed{~ a_n = a_{n-1} + p(n) -p(n-1) +1 \qquad n \ge 2 \qquad a_1 = 2 ~}\\ \begin{array}{rclcl} a_1 && &=& 2 \\ a_2 &=& a_1 + p(2)-p(1) + 1 = 2 + p(2)-2 + 1 &=& p(2) + 1\\ a_3 &=& a_2 + p(3)-p(2) + 1 = (~p(2) + 1~) + p(3)-p(2) + 1 &=& p(3) + 2 \\ a_4 &=& a_3 + p(4)-p(3) + 1 = (~p(3) + 2~) + p(4)-p(3) + 1 &=& p(4) + 3 \\ a_5 &=& a_4 + p(5)-p(4) + 1 = (~p(4) + 3~) + p(5)-p(4) + 1 &=& p(5) + 4\\ \cdots \\ a_n &=& a_{n-1} + p(n)-p(n-1) + 1 = (~p(n-1)+ n-2 ~)+ p(n)-p(n-1) + 1 &=& p(n) + n - 1\\ \end{array}\\ \boxed{~ a_n = n + p(n) -1 ~}\)

 

laugh

heureka  Jan 15, 2016
edited by heureka  Jan 15, 2016
edited by heureka  Jan 15, 2016
Sort: 

5+0 Answers

 #1
avatar+18712 
+15

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

 

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

 

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

 

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

 

laugh

heureka  Jan 14, 2016
 #2
avatar+90988 
+5

Thanks Heureka,

Here is a site that you will like guest :)

It is the

Online Encyclopeadia of Integer Sequences  :)

 

https://oeis.org/search?q=2%2C+4%2C+7%2C+10%2C+15%2C+18%2C+23%2C+26&sort=&language=english&go=Search

Melody  Jan 14, 2016
 #3
avatar
+5

Thanks heureka: brilliant work!. In my book it is stated slightly different: It is the sequence of natual or counting numbers added to prime numbers - 1. So, we have:

Counting numbers: 1, 2, 3, 4, 5, 6............

Prime numbers    + 2, 3, 5, 7, 11, 13...............

Subtract 1  from      2, 4, 7, 10, 15, 18..........etc.

each term

Guest Jan 14, 2016
 #4
avatar+78577 
0

That really IS brilliant, Heureka....!!!!!

 

 

cool cool cool

CPhill  Jan 14, 2016
 #5
avatar+18712 
+15
Best Answer

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)

 

Thanks heureka: brilliant work!. In my book it is stated slightly different: It is the sequence of natual or counting numbers added to prime numbers - 1. So, we have:

Counting numbers: 1, 2, 3, 4, 5, 6............

Prime numbers    + 2, 3, 5, 7, 11, 13...............

Subtract 1  from      2, 4, 7, 10, 15, 18..........etc.

each term

 

\(\text{The formula is}\\ \boxed{~ a_n = a_{n-1} + p(n) -p(n-1) +1 \qquad n \ge 2 \qquad a_1 = 2 ~}\\ \begin{array}{rclcl} a_1 && &=& 2 \\ a_2 &=& a_1 + p(2)-p(1) + 1 = 2 + p(2)-2 + 1 &=& p(2) + 1\\ a_3 &=& a_2 + p(3)-p(2) + 1 = (~p(2) + 1~) + p(3)-p(2) + 1 &=& p(3) + 2 \\ a_4 &=& a_3 + p(4)-p(3) + 1 = (~p(3) + 2~) + p(4)-p(3) + 1 &=& p(4) + 3 \\ a_5 &=& a_4 + p(5)-p(4) + 1 = (~p(4) + 3~) + p(5)-p(4) + 1 &=& p(5) + 4\\ \cdots \\ a_n &=& a_{n-1} + p(n)-p(n-1) + 1 = (~p(n-1)+ n-2 ~)+ p(n)-p(n-1) + 1 &=& p(n) + n - 1\\ \end{array}\\ \boxed{~ a_n = n + p(n) -1 ~}\)

 

laugh

heureka  Jan 15, 2016
edited by heureka  Jan 15, 2016
edited by heureka  Jan 15, 2016

10 Online Users

avatar
We use cookies to personalise content and ads, to provide social media features and to analyse our traffic. We also share information about your use of our site with our social media, advertising and analytics partners.  See details