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Question Two separate figures are given below. Each figure shows the first few positions in a sequence of arrangements made with sticks. Identify the pattern and answer the following questions for each figure:
(a) How many squares are in position number 11 of the sequence?
(b) How many sticks are needed to arrange position number 11 of the sequence?
(c) Can an arrangement in this sequence be made using exactly 85 sticks? If yes, which position number will it correspond to?
(d) Can an arrangement in this
So, no. of sticks needed in 11th position will be 13 + 9 × 10 = 103 (c) Let the nth arrangement have 85 sticks. 13 + (n – 1) × 9 = 85 ⇒ 13 + 9n – 9 = 85 ⇒ 4 + 9n = 85 ⇒ 9n = 85 – 4 ⇒ 9n = 81 ⇒ n = 9 Yes, the arrangement will have 85 sticks. (d) Let the nth arrangement have 150 sticks. 13 + (n – 1) × 9 = 150 ⇒ 13 + 9n – 9 = 150 ⇒ 4 + 9n = 150 ⇒ 9n = 150 – 4 ⇒ 9n = 146 ⇒ n = \(\frac {146}{9}\) ⇒ n = 16\(\frac {2}{9}\) It is not a whole number. So, no arrangement can be made with 150 sticks.