# Math

posted by K

Use mathematical induction to prove the property for all positive integers n.

A factor of (n^3+3n^2+2n) is 3.

1. Steve

1^3+3*1^2+2*1 = 1+3+2 = 6 = 3*2
so, true for n=1

so, assuming it's true for n=k, work with n=k+1:

(k+1)^3 + 3(k+1)^2 + 2(k+1)
= k^3+3k^2+3k+1 + 3k^2+6k+3 + 2k+2
= k^3+3k^2+2k + 3k+1 + 3k^2+6k+3 + 2
we know that k^3+3k^2+2k is a multiple of 3, so check out the rest. Rearranging things, we have

3k^2+9k+3

This is clearly also a multiple of 3.

So,
P(1)
P(k) => P(k+1)
so P is true for all k.

## Similar Questions

1. ### Calculus

Use mathematical induction to prove that each proposition is valid for all positive integral values of n. 5^n + 3 is divisible by 4.
2. ### Math - Mathematical Induction

3. Prove by induction that∑_(r=1)^n▒〖r(r+4)=1/6 n(n+1)(2n+13)〗. 5. It is given that u_1=1 and u_(n+1)=3u_n+2n-2 where n is a positive integer. Prove, by induction, that u_n=3^n/2-n+1/2. 14. The rth term of …
3. ### AP Calc

Use mathematical induction to prove that the statement holds for all positive integers. Also, can you label the basis, hypothesis, and induction step in each problem. Thanks 1. 2+4+6+...+2n=n^2+n 2. 8+10+12+...+(2n+6)=n^2+7n
4. ### Calculus

Use mathematical induction to prove that the statement holds for all positive integers. Also, label the basis, hypothesis, and induction step. 1 + 5 + 9 + … + (4n -3)= n(2n-1)
5. ### Math

Use mathematical induction to prove that 2^(3n) - 3^n is divisible by 5 for all positive integers. ThankS!
6. ### pre-calc

Use mathematical induction to prove that the statement is true for every positive integer n. Show your work. 2 is a factor of n2 -n+2
7. ### pre calc

Use mathematical induction to prove that the statement is true for every positive integer n. Show your work. 2 is a factor of n2 - n + 2
8. ### pre calc

Use mathematical induction to prove that the statement is true for every positive integer n. Show your work. 2 is a factor of n2 - n + 2
9. ### math

Use mathematical induction to prove that for all integers n ≥ 5, 1 + 4n < 2n
10. ### Mathematical Induction

Use mathematical induction to prove that the following is true. 8+11+14...+(3n+5)=1/2n(3n+13), for all n in the set of natural numbers.

More Similar Questions