1. **Problem statement:** (i) Given the pattern ৮, ১৩, ১৮, ২৩, ... or numerically 8, 13, 18, 23, ... (ii) The algebraic sequence ৫x + ২ or 5x + 2. (a) Draw the 3rd and 4th terms of the sequence ৪x + ১ or 4x + 1. (b) Explain which algebraic sequence the pattern (i) supports. (c) Find how many squares are needed for the first 11 terms of (ii) 5x + 2. 2. For a rectangular field with 3 processors and grass laying cost of 1822.50 at rate 7.50 per meter: (a) Find the length of the rectangular field using the processor measure 8 meters. (b) Find the length and width of the rectangular field. (c) Find how many 25 meter border girders are needed to make a square field of equal area. --- 2. **Step-by-step solutions:** **(1)(a) Find 3rd and 4th terms of 4x + 1:** For x=3: $$4(3) + 1 = 12 + 1 = 13$$ For x=4: $$4(4) + 1 = 16 + 1 = 17$$ So, 3rd term is 13 and 4th term is 17. **(1)(b) Identify the sequence pattern matching 8, 13, 18, 23,...** The pattern increases by 5 each time (common difference = 5), so it is an arithmetic sequence. General term: $$a_n = a_1 + (n - 1)d = 8 + (n-1)5 = 5n + 3$$ Compare with given sequence 5x + 2; ours is 5n + 3, close but different by 1. The pattern fits better to $$5n + 3$$. **(1)(c) Number of squares needed for first 11 terms of 5x + 2:** This sequence: $$a_n = 5n + 2$$ Sum of first 11 terms: $$S_{11} = \sum_{n=1}^{11} (5n + 2) = 5 \sum_{n=1}^{11} n + 2 \times 11 = 5 \times \frac{11 \times 12}{2} + 22 = 5 \times 66 + 22 = 330 + 22 = 352$$ So, 352 squares are needed. **(2)(a) Find length using 3 processors of 8 m each:** Total length = number of processors × length per processor = $$3 \times 8 = 24$$ meters. **(2)(b) Find length and width given total cost 1822.50 and cost rate 7.50 per meter:** Total cost = perimeter × cost per meter Perimeter $$P = \frac{1822.50}{7.50} = 243$$ meters. We know length $$L = 24$$ meters. Perimeter $$P = 2(L + W)$$ so, $$243 = 2(24 + W) \Rightarrow 24 + W = 121.5 \Rightarrow W = 121.5 - 24 = 97.5$$ Width $$W = 97.5$$ meters. **(2)(c) Find number of 25m border girders to build square field with equal area:** Area of rectangular field $$A = L \times W = 24 \times 97.5 = 2340$$ m² Side of square field $$s = \sqrt{2340} \approx 48.37$$ meters. Number of 25 m girders per side: $$\text{girders per side} = \lceil \frac{48.37}{25} \rceil = 2$$ Total girders for 4 sides: $$4 \times 2 = 8$$ --- **Final answers:** (a) 3rd term = 13, 4th term = 17 (b) Pattern fits arithmetic sequence $$5n + 3$$ (c) Squares needed for first 11 terms = 352 (2)(a) Length = 24 m (2)(b) Width = 97.5 m (2)(c) Number of girders needed = 8