# AP Calc

posted by .

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

• AP Calc -

assume true for n=k. Then when n=k+1, we have

2+4+...+2k+(2k+2) = k^2 + k + 2k+2
= k^2 + 2k + 1 + k + 1
= (k+1)^2 + (k+1)

Since true for n=1, true for n=2,3,4...

Similarly,

8+10+...+(2k+6)+(2k+8) = k^2 + 7k + (2k+8)
= k^2 + 2k + 1 + 7k + 7
= (k+1)^2 + 7(k+1)

## Similar Questions

1. ### Calculus

Write the statements for the basis, the induction hypothesis, and the induction step for the questions below. 1. 1 + 2 +…+n = n(n+1)/2 2. 1/1*2 + 1/2*3+…+1/n*(n+1) = n/n+1
2. ### 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)
3. ### Math

Use mathematical induction to prove that 2^(3n) - 3^n is divisible by 5 for all positive integers. ThankS!
4. ### Mathematical induction. I'm stuck. So far I have..

For all integers n ≥ 1, prove the following statement using mathematical induction. 1+2^1 +2^2 +...+2^n = 2^(n+1) −1 Here's what I have so far 1. Prove the base step let n=1 2^1=2^(1+1)-1 False. Someone else suggested that …
5. ### 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
6. ### 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 …
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