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use mathematical induction to show that (a) 1 + 4 +7 + ...+(3n -2) = n (3n - 1)/2 for n is the element of natural numbers, n is greater than or equal to 1.(

User Magggi
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1 Answer

5 votes

Answer:

  • It is true for n=1
  • if n=k is true then n=k+1 is also true

Explanation:

  • n=1:


1=1(3*1-1)/2\\1=2/2\\1=1\\ TRUE

  • n=k:


1 + 4 +7 + ...+(3k -2) = k (3k - 1)/2\\

  • n=k+1:


1 + 4 +7 + ...+(3k -2)+ (3(k+1)-2) =k(3(k+1)-1)/2 \\

We replace the first part of the equation with our value for n=k:


k (3k - 1)/2+(3(k+1)-2)=k(3(k+1)-1)/2 \\

we develop both sides of the equation to verify equality:


3k^(2) /2-k/2+3k+1=(k+1)(3k+2)/2\\\\3k^(2) /2+5k/2+1=(3k^(2)+5k+2)/2\\\\3k^(2) /2+5k/2+1=3k^(2) /2+5k/2+1 TRUE

User EMC
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