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If you're going on to Calculus, you're going to need these!

Here's the first one:

the summation of ( ak + bk ) as k goes from 1 to n = the summation of ( ak ) as k goes from 1 to n + the summation of ( bk ) as k goes from 1 to n

Here's an example so you can believe it:

Find this sum:

the summation of ( 2k + k ) as k goes from 1 to 3

Now, find these sums and add them:

the summation of ( 2k ) as k goes from 1 to 3 + the summation of ( k ) as k goes from 1 to 3

Do you believe it?  (You should have gotten the same thing for both.)

This may not seem like a big deal, but you'll need it to prove some important calculus stuff, so let's get used to it.

Here's the proof:

the summation of ( ak + bk ) as k goes from 1 to n Just expand it out...

= ( a1 + b1 ) + ( a2 + b2 ) + ( a3 + b3 ) + ... + ( an + bn )

Now, use the commutative and associative properties...  Gather up the a's...  Gather up the b's...

= ( a1 + a2 + a3 + ... + an ) + ( b1 + b2 + b3 + ... + bn ) = the summation of ( ak ) as k goes from 1 to n + the summation of ( bk ) as k goes from 1 to n ... black square ... this square means the proof is done!

Here are the other two properties -- I'll let you do the proofs:

the summation of ( ak - bk ) as k goes from 1 to n = the summation of ( ak ) as k goes from 1 to n - the summation of ( bk ) as k goes from 1 to n

the summation of ( c * ak ) as k goes from 1 to n = c * the summation of ( ak ) as k goes from 1 to n

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