Question

USE THE PRINCIPLE OF MATHEMATICAL INDUCTION TO PROVE THE FOLLOWING STATEMENT FOR ALL n∈N

1²+3²+5²...+(2n-1)²=1/3n (2n-1) (2n+1)
...
do not give improper answers otherwise i can report

Answer 1

Answer:

Let P(n) be the given statement, i.e., P(n) : (2n + 1) < 2n

for all natural

numbers, n ≥ 3. We observe that P(3) is true, since

2.3 + 1 = 7 < 8 = 23

Assume that P(n) is true for some natural number k, i.e., 2k + 1 < 2k

To prove P(k + 1) is true, we have to show that 2(k + 1) + 1 < 2k+1. Now, we have

2(k + 1) + 1 = 2 k + 3

= 2k + 1 + 2 < 2k + 2 < 2k

. 2 = 2k + 1

.

Thus P(k + 1) is true, whenever P(k) is true.

Hence, by the Principle of Mathematical Induction P(n) is true for all natural

numbers, n ≥ 3

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