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Use mathematical induction to prove 6 is a factor of n^3 + 3n^2 + 2n. Please pls pls pls help me. Thank you.
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1. According to the Method of Mathematical induction, check the statement at n = 1:
1^3 + 3*1^2 + 2^1 = 1 + 3 + 2 = 6
and the statement is TRUE.
2. According to the Method of Mathematical induction, let us assume that the statement is true for n= k, i.e. let assume that
is a multiple of 6.
Consider the polynomial expression at n = k+1. You have
= = regroup the terms = + . (1)
According to the induction assumption, the term is a multiple of 6.
The last term = = 3*(k+1)*(k+2) is the thrice the product of two consecutive integer numbers.
So, this product is multiple of 2, and when multiplied by 3, is a multiple of 6.
Thus, the right side of (1) is a multiple of 6, and the induction step is proved.
3. According to the principle of the Mathematical induction, the original statement is proved.
QED.