Question

Which of the following can be the square of a natural number ‘n’?
(i) sum of the squares of first n natural numbers
(ii) sum of the first n natural numbers
(iii) sum of first (n – 1) natural numbers
(iv) sum of first ‘n’ odd natural numbers.

Answer 1

Answer:

ANSWER 1 sum of the square of first n natural number

Answer 2

Which of the following can be the square of a natural number ‘n’?

(i) sum of the squares of first n natural numbers

$S_{n} = 1^{2} + 2^{2} + 3^{2} + \cdot\cdot \cdot + n^{2} \\= \frac{n(n+1)(2n+1)}{6}$

(ii) sum of the first n natural numbers

$S_{n} = 1+2+3+\cdot\cdot\cdot + n \\= \frac{n(n+1)}{2}$

(iii) sum of first (n – 1) natural numbers

$S_{n} = 1 + 2 + 3 + \cdot \cdot \cdot + (n-1) \\= \frac{(n-1)(n-1+1)}{2} \\= \frac{(n-1)n}{2}$

(iv) sum of first ‘n’ odd natural numbers.

$\pink { S_{n}= 1+3+5+\cdot \cdot \cdot + n\:terms }\\ \green { = n^{2}}$

Therefore.,

$Option \: \pink { (iv) } \: is \: correct .$

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