Question

Use the principle of induction to prove that

$1 + 3 + 5 + ... + (2n - 1) = n^{2}$ for all n $\in \mathbb{N}$.

Answer 1

solve with the help of this one


Let us write

P(n): 1 + 3 + 5 + 7 + ... + (2n – 1) = n2

.

We wish to prove that P(n) is true for all n.

The first step in a proof that uses mathematical induction is to prove that

P (1) is true. This step is called the basic step. Obviously

1 = 12

, i.e., P(1) is true.

The next step is called the inductive step. Here, we suppose that P (k) is true for some

positive integer k and we need to prove that P (k + 1) is true. Since P (k) is true, we

have

1 + 3 + 5 + 7 + ... + (2k – 1) = k2 ... (1)

Consider

1 + 3 + 5 + 7 + ... + (2k – 1) + {2(k +1) – 1} ... (2)

= k2 + (2k + 1) = (k + 1)2 [Using (1)]

Therefore, P (k + 1) is true and the inductive proof is now completed.

Hence P(n) is true for all natural numbers n.