Mathematical induction proofs for sums and divisibility

[evaboo]

hwk helpp please helpp! thanks in advance

"Prove each of the following using Mathematical induction;" show all steps
pleasee someone help.. i have a test on this tommorow and i just need some examples.. could you also try to show all steps including the words so i understand how you got there? thakns so much in advance~!

1. -1/2, -1/4, -1/8... -1/2^n = (1/(2^n))-1

2. a + (a+d)+(a+2d)+...+[a+(n-1)d] = (n/2)[2a+(n-1)d]

3. 1^3 + 2^3 + 3^3... + n^3 = (n^2(n+1)^2)/(4)

4. show that (3^(4n))-1 is dividislbe by 80 for all positive integral values of n

 

[dextercioby]

Well,did u read the HW section guidelines ?We don't do hw-s here,we only help people do their homeworks,if they get stuck.

But u haven't even started.How about some ideas...?

Daniel.

EDIT:Please,DO NOT DOUBLE POST !

 

[thepatientmental]

Hello, let me help you with these problems using mathematical induction.

1. Base Case: For n = 1, we have -1/2 = (1/2) - 1 = -1/2. This is true.

2. Inductive Hypothesis: Assume that the formula holds for some positive integer k.

3. Inductive Step: We need to show that the formula holds for k+1.

a + (a+d)+(a+2d)+...+[a+(k-1)d]+[a+kd] = (k/2)[2a+(k-1)d]+[a+kd] (by inductive hypothesis)
= (k/2)[2a+kd] (since [a+kd] = 2a+kd)
= (k/2)[2a+(k+1)d]
= [(k+1)/2][2a+(k+1)d]

Therefore, the formula holds for k+1, and by mathematical induction, it holds for all positive integers n.

3. Base Case: For n = 1, we have 1^3 = (1^2(1+1)^2)/(4) = 1. This is true.

4. Inductive Hypothesis: Assume that the formula holds for some positive integer k.

5. Inductive Step: We need to show that the formula holds for k+1.

1^3 + 2^3 + 3^3... + k^3 + (k+1)^3 = (k^2(k+1)^2)/4 + (k+1)^3 (by inductive hypothesis)
= (k^4 + 2k^3 + k^2)/4 + (k+1)^3
= (k^4 + 2k^3 + k^2 + 4k^3 + 12k^2 + 12k + 4)/4 (by expanding (k+1)^3)
= (k+1)^2(k+2)^2/4

Therefore, the formula holds for k+1, and by mathematical induction, it holds for all positive integers n.

4. Base Case: For n = 1, we have (3^(4*1))-1 = 80 which is divisible by 80. This is true.

5. Inductive Hypothesis: Assume that the formula