Question

if the sum of first n even natural numbers is 420 find the value of n

Answer 1

420 = n (n + 1 )      use distributive property
400 + 20 = n2 + n     transpose
400 - n2 = n - 20     get the square root of 400 - n2 then it becomes
20 - n = n - 20        transpose
20 + 20 = n + n       then add
40 = 2n         then divide by 2 on each side to get the value of n
n = 20
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Answer 2

$\text{Sum of first n natural numbers is given as:}\\ \\ S_n = \frac{n(n + 1)}{2} \\ \\\text{Sum of first n natural even numbers is given as:}\\ \\S_{n(even)} = 2 \times \frac{n(n + 1)}{2} = n(n + 1) \\ \\ n(n + 1) = 420 \\ \\ {n}^{2} + n = 420 \\ \\ {n}^{2} + n - 420 = 0 \\ \\ {n}^{2} + 21 n - 20n - 420 = 0 \\ \\ n(n + 21) - 20(n + 21) = 0 \\ \\ (n + 21)(n - 20) = 0 \\ \\ n = 20 \: \: and \: \: - 21 \\ \\ n \: can \: not \: be \: negative \\ \\ n = 20$