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Mathematical induction question in Discrete Math.?
First time doing mathematical induction. Any help is appreciated.
Use mathematical induction to give a careful proof that
2+8+14+....+(6n+2)=(n+1)(3n+2) for all intergers n>=1.
1 Answer
- dennisLv 69 years agoFavorite Answer
Should be for all integers n >= 0
First we have to prove that the statement is true for n = 0
LHS = 2 and RHS = (0+1)(0+2) = 2 so true for n= 0
Now assume statement is true for n = k
so 2+8+14+....+(6k+2)=(k+1)(3k+2) .......(1)
and we have to show that the statement is true for n = k+1 which is
2 + 8 + 14 + .....(6k+2) + ( 6(k+1) +2) = ((k+1)+1)(3(k+1) +2)
To do this we add 6(k+1) +2 = 6k + 8 to both sides of (1) to get
2+8+14+....+(6k+2) +(6k+8) =(k+1)(3k+2) +6k +8
= 3k^2 + 11k + 10 = (k+2)( 3k+5)=((k+1)+1)(3(k+1) +2).
So the statement is true for n = k+1, so by induction it is true for all integers n >=0.
