prove by mathematical induction that 1³+2³+3³+.......+n³=(n (n+1)/2)²
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Given:
1³+2³+3³+...+n³=(n (n+1)/2)²
To Find:
Proven by mathematical induction
Solution:
First, we should know the process of mathematical induction, for an equation P(n)
- The equation should be true for n=1
- If the equation is true for n=k then it is also true for n=k+1
Then equation P(n) will be true for all natural numbers and the equation will be proven by mathematical induction
So, for n=1, we have
hence it is true for n=1
Now for n=k, it is true,
so it needs to be true for n=k+1 also, we have,
Hence it is also true for n=k+1, so p(n) is true for all natural numbers.
Hence, Proved by mathematical induction that 1³+2³+3³+...+n³=(n (n+1)/2)².
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