SOLUTION: Please help with this questions, prove by induction that the formulas positive integral n. 1) 1+4+7+...+(3n-2)= ½n(3n-1) 2) 1+5+9+...+(4n-3)=n(2n-1) 3) 5+10+15+...+5n=5/2n(n+1)

Question 1190618: Please help with this questions, prove by induction that the formulas positive integral n.
1) 1+4+7+...+(3n-2)= ½n(3n-1)
2) 1+5+9+...+(4n-3)=n(2n-1)
3) 5+10+15+...+5n=5/2n(n+1)
4) 1²+3²+5²+....+(2n-1)²=1/3n(4n²-1)
5) 1+2+4+8+....+2^n-1=2^n-1
Answer by ikleyn(52780)   (Show Source): You can put this solution on YOUR website!
.

Tasks 1), 2) and 3) relate to arithmetic progressions.

On Method of mathematical induction for arithmetic progressions, see the lesson
    - Mathematical induction and arithmetic progressions
in this site.

Find there the solutions for similar problems.

Task 4) relates to the sum of squares of natural numbers.

On Method of mathematical induction for similar sequence, see the lesson
    - Mathematical induction for sequences other than arithmetic or geometric
in this site.

Task 5) relates to the sum of terms of geometric progression.

On Method of mathematical induction for geometric progression, see the lesson
    - Mathematical induction and geometric progressions
in this site.

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".

Save the link to this textbook together with its description

Free of charge online textbook in ALGEBRA-II
https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson

into your archive and use when it is needed.

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