Prove by the principle of mathematical induction that 1+2+3+...+n<(2n+

1.	Prove by the principle of mathematical induction that 1+2+3+...+n<(2n+1)28, for all n∈N. Or Using the principle of mathematical induction, prove that 1.3+2.32+3.33+...+n.3n=(2n−1)3n+1+34. for all n∈N.
Answer» Prove by the principle of mathematical induction that $1 + 2 + 3 + . . . + n < \frac{(2 n + 1)^{2}}{8}$ , for all $n \in N .$ Or Using the principle of mathematical induction, prove that $1.3 + {2.3}^{2} + {3.3}^{3} + . . . + n {.3}^{n} = \frac{(2 n - 1) 3^{n + 1} + 3}{4} .$ for all $n \in N .$

Prove by the principle of mathematical induction that 1+2+3+...+n<(2n+1)28, for all n∈N. Or Using the principle of mathematical induction, prove that 1.3+2.32+3.33+...+n.3n=(2n−1)3n+1+34. for all n∈N.

Answer»

Prove by the principle of mathematical induction that $1 + 2 + 3 + . . . + n < \frac{(2 n + 1)^{2}}{8}$ , for all $n \in N .$

Using the principle of mathematical induction, prove that

$1.3 + {2.3}^{2} + {3.3}^{3} + . . . + n {.3}^{n} = \frac{(2 n - 1) 3^{n + 1} + 3}{4} .$ for all $n \in N .$