pre-calculus

posted by .

proving that the statements are true for all natural numbers

1+2+3+....+n=n(n+1)/2

i don't understand when trying to prove that P(k+1) = 1+2+3+...+k+k(k+1)= 1/2

how do you do this step???

  • pre-calculus -

    This is called induction. You assume it's true for n=k. Then, if it's true for k, you show that it must also be true for k+1.

    Then, show it's true for n=1.
    That means it's true for n=2, thus 3, and so on.

    So, assume that
    1+2+3+...+k = k(k+1)/2
    Now consider
    1+2+3+...+k+k+1
    Assuming our hypothesis is tur for n=k, that means that
    1+2+3+...+k+1 = k(k+1)/2 + k+1
    This is the crucial step. We already "know" what 1+2+3+...+k is, so we just substitute it in.

    Now start rearranging the right hand side:

    k(k+1)/2 + k+1 = [k(k+1) + 2(k+1)]/2
    = (k+1)(k+2)/2
    Wow! This is just our formula, if we substitute in k+1 for k.

    If 1+2+3+...+k = f(k)
    then 1+2+3+...+k+k+1 = f(k+1)

    Now, what if n=1?
    1 = 1(1+1)/2 = 1(2)/2 = 1
    so our formula is true for n=1.
    We have shown that it must also be true for n=2, n=3, ...

Respond to this Question

First Name
School Subject
Your Answer

Similar Questions

  1. proof by mathematical induction

    subject is PreCalulus. 2^(k+3) = and < (k+3)! i know how to do proving using mathematical induction when its just an equal sign, but I don't understand what to do when its an inequality. thank you!!! Here is what I would do Step …
  2. Math

    Use mathematical induction to prove that 5^(n) - 1 is divisible by four for all natural numbers n. Hint: if a number is divisible by 4, then it has a factor of 4. also, -1 = -5 +4 This is a take home test so I don't want the answer …
  3. algebra

    Are the following true or false? 1. The difference of two natural numbers is always a natural number. 2. The sum of two natural numbers is always a natural number. 3. The quotient of two natural numbers is always a natural number.
  4. pre calc

    show that the statements are true for all natural numbers 1^3 +2^3 +5^3+....+(2n+3)^3 =n^2(2n^2 -1)
  5. Math pre-calculus

    Write an expression for all of the vertical asymptotes of y=-3 csc(pi/4 *x) Can you list step by step to show how to do it , I don't understand this problem !!! Thanks
  6. Pre-Calculus/Trigonometry

    Help Me, Please Here is a table of values for y = f(x). x: 0, 5, 10, 15, 20, 25, 30, 35, 40 f(x): 5, 6, 7,8, 9, 10, 11, 12, 13 Mark the statements that are true. A. f(5) = 6 B. The domain for f(x) is the set {5, 6, 7, 8, 10, 11, 12, …
  7. Algebra ASAP

    so this is a fill in on a worksheet and I am having difficulty as the ones I inserted are incorrect can anybody help me how to do it all, sorry it's a long problem. Show that 3^2n − 1 is divisible by 8 for all natural numbers …
  8. Calculus

    If the functions f and g are defined for all real numbers and f is an antiderative of g, which of the following statements is NOT necessarily true?
  9. Calculus

    If the functions f and g are defined for all real numbers and f is an antiderivative of g, which of the following statements is NOT necessarily true. (A)If g(x)>0 for all x, then f is increasing (B)If g(a)=0, then f(x) has a horizontal …
  10. Math

    True or false (check) A. All integers are natural numbers FALSE B. All rational numbers are integers TRUE C. All natural numbers are whole numbers TRUE D. All rational numbers are roots FALSE

More Similar Questions