SOLUTION: Use mathematical induction to prove the formula. formula: https://s11.postimg.org/ibmz0jczn/math16.png

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Question 1081848: Use mathematical induction to prove the formula.
formula: https://s11.postimg.org/ibmz0jczn/math16.png
Found 2 solutions by math_helper, ikleyn:
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!
+S%5B1%5D+=+3++
+S%5B2%5D+=+9+=+3%2A%282%29%2A%282%2B1%29%2F2+
…
Assume +S%5Bk%5D+=+3%2Ak%28k%2B1%29%2F2+ (*)
What is +S%5Bk%2B1%5D+ ?
It is the previous sum plus 3*(k+1):
+S%5Bk%2B1%5D+=+S%5Bk%5D+%2B+3%28k%2B1%29+
+S%5Bk%2B1%5D+=+3k%28k%2B1%29%2F2+%2B+3%28k%2B1%29+
+S%5Bk%2B1%5D+=+3%28k%2B1%29%2A%28k%2F2+%2B+1%29
+S%5Bk%2B1%5D+=+3%28k%2B1%29%2A%28k%2B2%29%2F2+ <<<— <<< DONE (same form as (*) )
—
If the above is not clear, let h=k+1, then +S%5Bk%2B1%5D+=+S%5Bh%5D+=+3h%28h%2B1%29%2F2+ which should make it clear.

Answer by ikleyn(52810) About Me  (Show Source):
You can put this solution on YOUR website!
.
On mathematical induction, see the lessons
    - Mathematical induction and arithmetic progressions
    - Mathematical induction and geometric progressions
    - Mathematical induction for sequences other than arithmetic or geometric
    - Proving inequalities by the method of Mathematical Induction

Also, you have this free of charge online textbook in ALGEBRA-II in this site
    - ALGEBRA-II - YOUR ONLINE TEXTBOOK.

The referred lesson is the part of this online textbook under the topic
"Method of Mathematical induction".