12-32-52-2 n-12-13 n4 n 2-1

Let P(n) be the given statement. Now, P(n)=12+32+52+...+(2n 1)2=13n(4n2 1)Step #160;1: #160; #160;P(1)=12=1=13 #215;1 #215;(4 1)Hence, #160;P(1) # ...

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Byju's Answer

Standard XII

Mathematics

Proof by mathematical induction

12+32+52+…+2 ...

Question

1² + 3² + 5² + ... + (2n − 1)² = $\frac{1}{3} n (4 n^{2} - 1)$

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Solution

Let P(n) be the given statement.
Now,
$P (n) = 1^{2} + 3^{2} + 5^{2} + . . . + (2 n - 1)^{2} = \frac{1}{3} n (4 n^{2} - 1) S t e p 1 : P (1) = 1^{2} = 1 = \frac{1}{3} \times 1 \times (4 - 1) Hence, P (1) is true . Step 2 : Let P (m) be true . T h e n, 1^{2} + 3^{2} + . . . + (2 m - 1)^{2} = \frac{1}{3} m (4 m^{2} - 1) T o prove : P (m + 1) is true whenever P (m) is true . T h a t i s, 1^{2} + 3^{2} = . . . + (2 m + 1)^{2} = \frac{1}{3} (m + 1) \{4 (m + 1)^{2} - 1\} W e k n o w t h a t P (m) is true . T h u s, w e h a v e : 1^{2} + 3^{2} + . . . + (2 m - 1)^{2} = \frac{1}{3} m (4 m^{2} - 1) \Rightarrow 1^{2} + 3^{2} + . . . + (2 m - 1)^{2} + (2 m + 1)^{2} = \frac{1}{3} m (4 m^{2} - 1) + (2 m + 1)^{2} [Adding (2 m + 1)^{2} to both sides] \Rightarrow P (m + 1) = \frac{1}{3} (4 m^{3} - m + 12 m^{2} + 12 m + 3) \Rightarrow P (m + 1) = \frac{1}{3} (4 m^{3} - m + 8 m^{2} + 4 m + 4 m^{2} + 8 m + 3) = \frac{1}{3} (m + 1) (4 m^{2} + 8 m + 3) = \frac{1}{3} (m + 1) (4 (m + 1)^{2} - 1) T h u s, P (m + 1) i s true . B y t h e p rinciple of m athematical i nduction, P (n) is true for all n \in N .$

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Similar questions

Prove $1^{2} + 3^{2} + 5^{2} + \dots + (2 n - 1)^{2} = \frac{n (2 n - 1) (2 n + 1)}{3}$

Q. Show that

1^{2} + 3^{2} + 5^{2} + . . . + {(2 n - 1)}^{2} = \frac{n (2 n - 1) (2 n + 1)}{3} .

Q. Prove that

1^{2} + 3^{2} + 5^{2} + . . . + {(2 n - 1)}^{2} = \frac{n}{3} (2 n - 1) (2 n + 1)

Q. Find sum

1^{2} + 3^{2} + 5^{2} + . . . + {(2 n - 1)}^{2}

Q. Prove by Mathematical induction that

1^{2} + 3^{2} + 5^{2} . . . {(2 n - 1)}^{2} = \frac{n (2 n - 1) (2 n + 1)}{3} \forall n \in N

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12 + 32 + 52 + ... + (2n − 1)2 = 13n(4n2-1)

1² + 3² + 5² + ... + (2n − 1)² = $\frac{1}{3} n (4 n^{2} - 1)$