how would u use mathematical induction to prove this statement.?

Question

n
∑ (ai + bi)= ∑ ai + ∑ bi
i=1

Answer 1

Start with n=1.
If n=1, both sides of the equation equal (a+b)
Now we need to show that if:
n n n
∑ (ai + bi)= ∑ ai + ∑ bi
i=1 i=1 i=1
Then
n+1 n+1 n+1
∑ (ai + bi)= ∑ ai + ∑ bi
i=1 i=1 i=1
Here's the proof of that:

n+1 n
∑ (ai + bi)= ∑ + (n+1)(a+b)
i=1 i=1
Which, from our first step is equal to: (I am substituting the summation and expanding (n+1)(a+b)

n n
∑ ai + ∑ bi + a(n+1) + b(n+1)
i=1 i=1
and this is just equal to: (I am putting the last two terms into their respective summations)

n+1 n+1
∑ ai + ∑ bi
i=1 i=1
And that's exactly what you needed!!
I'm assuming for this that you understand induction... if you don't (or need me to explain any of the steps) just write a message in the additional details.

For some reason, when I post this the n and n+1 terms don't appear over the summations properly... neither do the i=1 terms...if you see a line of a bunch of n's or n+1's just pretend they're on top of their respective summations (and do the same for the i=1's below the summations)

Answer 2

I think its one of those "hold down alt and press numbers" symbols...But I could be wrong.

Answer 3

Solve for a few experimental values of n - if it works for these test values, we can postulate that it works for any value of n.

Answer 4

first how did u type that character ?∑