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Good morning maam/sir..kindly help me with the following:
Derive the formula representing the following sums:
1. (1/1.2) + (1/2.3) + (1/3.4) +...+ 1/n.(n+1)
2. (n+1)^2 + (n+2)^2 + (n+3)^2 +...+ (2n)^2
3. 1.n + 2(n-1) + 3(n-2) +...+ (n-2).2 + n.1
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1. = .
See the lesson Mathematical induction for sequences other than arithmetic or geometric, Problem 4.
2. =
= -
= .
It uses the basic formula = .
See the lesson Mathematical induction for sequences other than arithmetic or geometric, Problem 1.
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".
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For #3 I will not derive the final formula.
I will simply explain you how to get it.
+ + =
+ + =
+ + =
. . . . . . . . . . . . . . . . . . . . . . .
+ + =
+ + =
------------------------------------------- Add all equations, both the left sides and the right sides. You will get
+ + = ,
where
is exactly your sum under the question, and
= .
At this point I will stop my explanations.
If you have a necessary background/training/education in Math, then these tips are enough and you will be able to complete it on your own.
If you do not have it, then any further explanations are useless.
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By the way, your formula in #3 is not exactly correct.
The correct formula is
3. 1.n + 2(n-1) + 3(n-2) +...+ (n-1).2 + n.1
Honestly, I don't like very much to work for those who is not able to write his/her formulas
(and, in general, the condition to the problem) correctly.