Math

posted by .

n^3+2n is the multiple of 3 prove it by math induction method

  • Math -

    step 1:
    test for n = 1
    1^3 + 2(1) = 3 , which is a multiple of 3

    step 2:
    assume it is true for n = k
    that is, k^3 + 3k is a multiple of 3, or it is divisible by 3

    step 3:
    show that it is also true for n = k+1
    that is, show that (k+1)^3 + 3k is a multiple of 3

    let's take the difference
    (k+1)^3 + 2(k+1) - (k^3 + 2k)
    = k^3 + 3k^2 + 3k + 1 + 2k + 2 - k^3 - 2k
    = 3k^2 + 3k + 3
    = 3(k^2 + k + 1)
    which is divisible by 3, (since 3 is a factor)

    so n^3 + 3n is always a multiple of 3

    the property I used is the following:
    if 2 numbers are divisible by the same number, then their difference is divisible by that same number

    e.g. 91 and 49 are both divisible by 7
    then 91-49 or 42 is also divisible by 7
    -- try it for other numbers.
    since we knew the second number, k^3 + 2k , was divisible by 3 and the result was divisible by 3, then the first number, (k+1)^3 + 2(k+1) has to be divisible by 3

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. math : induction

    The reversal of a string w, denoted w^R, is the string "spelled backwards". For example (reverse)^R = esrever. A careful definition can be given by induction on the length of a string: 1. If w is a string of length 0, then w^R = epsilon …
  2. math

    Can anyone please help me with the following question: Prove by mathematical induction that 6^n + 4 is a multiple of 5, for nEN.
  3. math

    Prove by mathematical induction that : E (3r-5)= 3n^2-7n /2 r=1
  4. 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 …
  5. 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
  6. 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)
  7. Computer proof

    Prove by induction on all positive integer k that if m is any ordinary nfa with k states, and m has fewer than k - 1 transitions, then there exists a state of m that is not reachable. Let N be the λ-NFA: "L" for "λ" b >(1) …
  8. math

    use mathematical induction to prove -1/2^n = 1/2^n - 1
  9. Math

    Prove, by mathematical induction, or otherwise, that 1*1!+2*2!+3*3!+…+n*n!=(n+1)!-1
  10. Math, Induction

    prove by induction that 3.7^(2n)+1 is divisible by 4

More Similar Questions