SOLUTION: Use Mathematical Induction to show that the following statement is true for all natural numbers n: {{{ 1^3+2^3+3^3+...+}}} {{{n^3 = n^2(n+1)^2/(4) }}}
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Question 1104281: Use Mathematical Induction to show that the following statement is true for all natural numbers n:
Found 3 solutions by stanbon, math_helper, ikleyn:
Answer by stanbon(75887) (Show Source): You can put this solution on YOUR website!
Use Mathematical Induction to show that the following statement is true for all natural numbers n: 1^3+2^3+3^3+...+ n^3 = n^2(n+1)^2/(4)
-------
Show it is true for n = 1::
1^3 = [1^2(1+1)^2]/4 = (1*2^2)/4
1 = 4/4
1 = 1
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Assume it is true for n = k,
1^3 + 2^3 + ... + k^3 = [k^2(k+1)^2]/4
------
Show it is true for n = k+1
1^3 + 2^3 + ..+ k^3 + (k+1)^3 = [k^2(k+1)^2]/4 + (k+1)^3
Factor to getl::
= (k+1)^2[k^2/4 + (k+1)]
= (k+1)^2[(k^2 + 4k+ 4)]/4
= (k+1)^2[((k+1)+1)^2]/4
----
So it is true for n = k+1.
Cheers,
Stan H.
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Answer by math_helper(2461) (Show Source): You can put this solution on YOUR website!
For n=2:
and
—
For n=3:
and
—
Assume it is true for n=k. That is + ... + (*)
—
Now let n=k+1:
+ ... +
Which, upon substituting (*), gets us to:
=
factoring:
=
=
QED.
(Assuming (*) is true for n=k leads to (*) being true for n=k+1)
Answer by ikleyn(52780) (Show Source): You can put this solution on YOUR website!
.
For this and other similar problems on the method of Mathematical induction see the lesson
- Mathematical induction for sequences other than arithmetic or geometric
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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