Prove by mathematical induction \[ \frac{1}{1 \times 5} + \frac{1}{5 \times 9} + \cdots + \frac{1}{(4n-3)(4n+1)} = \frac{n}{4n+1} \] for all natural numbers n > 0.
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Prove by mathematical induction \[ 9^{n} + 3 \] is divisible by 4 for all natural numbers \( n \gt 0 \).
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Prove by mathematical induction \[ 5^{2n} - 1 \] is divisible by 24 for all natural numbers \( n \gt 0 \).
Solve
Prove by mathematical induction \[ \sum_{i=1}^n (2i-1) = n^2 \] for all natural numbers n > 0.
Solve