Question

Observe the following pattern in which each small square represents a unit square (square of side 1 unit).

Fig (i)	Fig (ii)	Fig (iii)

If the sum of number of unit squares in the n^th figure and (n + 2)^th figure is 290, find the value of n.

Answer 1

The pattern is 1², 2², 3², ... n², (n + 2)²

According to the question,

n² + (n + 2)² = 290

⇒ n² + n² + 4n + 4 = 290

⇒ 2n² + 4n + 4 = 290

⇒ 2(n² + 2n + 2) = 290

⇒ n² + 2n + 2 = 145

⇒ n² + 2n – 143 = 0

a = 1, b = 2, c = –143

`n = (-b ± sqrt(b^2 - 4ac))/(2a)`

= `(-2 ± sqrt(4 + 4 xx 143))/2`

⇒ `n = (-2 ± sqrt(576))/2`

⇒ `n = (-2 ± 24)/2`

⇒ `n = (-2 + 24)/2, (-2 - 24)/2`

⇒ `n = 22/2, (-26)/2`

⇒ n = 11, –13 (not valid as n must be positive)