Use mathematical induction to solve for all positive integers n.
A factor of $\left( n ^ { 3 } + 7 n ^ { 2 } + 6 n \right)$ is:

A)7
B)9
C)8
D)6
E)3

Question

Use mathematical induction to solve for all positive integers n. ​
A factor of $\left( n ^ { 3 } + 7 n ^ { 2 } + 6 n \right)$ is:
​
A)7
B)9
C)8
D)6
E)3

Quizplus · Accepted Answer

To prove by induction that 3 is a factor, check the base case n=1: $1^3 + 7(1)^2 + 6(1) = 14$, which is divisible by 3. Assume true for n=k, i.e., $k^3 + 7k^2 + 6k$ is divisible by 3. For n=k+1, $(k+1)^3 + 7(k+1)^2 + 6(k+1)$ simplifies to $k^3 + 3k^2 + 3k + 1 + 7k^2 + 14k + 7 + 6k + 6$, which is $k^3 + 7k^2 + 6k + 3(k^2 + 3k + 3)$. By the induction hypothesis, $k^3 + 7k^2 + 6k$ is divisible by 3, and so is $3(k^2 + 3k + 3)$. Thus, the expression is divisible by 3 for all positive integers n.

Use Mathematical Induction to Solve for All Positive Integers N (n3+7n2+6n)\left( n ^ { 3 } + 7 n ^ { 2 } + 6 n \right)(n3+7n2+6n)

Use Mathematical Induction to Solve for All Positive Integers N $\left( n ^ { 3 } + 7 n ^ { 2 } + 6 n \right)$