Question

Express 196 as sum of consecutive odd natural numbers.​

Answer 1

Answer:

a.

The sum of first n consecutive odd numbers is n².

Here 196 = 14².

∴ 196 is the sum of first 14 odd numbers.

nth odd number = 2n - 1

∴ 14th odd number = 2 x 14 - 1 = 28 - 1 = 27.

∴ 1 + 3 + 5 + ...... + 25 + 27 = 196

b.

If a² + b² = c², then a = m² - n², b = 2mn and c = m² + n².

We can consider 18 as b = 2mn.

2mn = 18

mn = 18 / 2 = 9

9 = 9 x 1

∴ m = 9 and n = 1 can be considered.

a = 9² - 1² = 81 - 1 = 80

b = 2 x 9 x 1 = 18

c = 9² + 1² = 81 + 1 = 82

∴ 80, 18 and 82 is a Pythagorean triplet.

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Answer 2

97+99=196

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