a) By selling an article at shs 22,500/= a shopkeeper makes loss of 10%. At what price must the shopkeeper sell the article in order to get a profit of 10%? b) Find the sum of the first four terms of

Here are the solutions to your questions: 3. MATHEMATICS a) By selling an article at shs 22,500/= a shopkeeper makes loss of 10%. At what price must the shopkeeper sell the article in order to get a profit of 10%? Step 1: Calculate the Cost Price (CP). A loss of 10% means the selling price is 90% of the cost price. 0.90 × CP = 22,500 CP = (22,500)/(0.90) CP = 25,000 shs Step 2: Calculate the selling price for a 10% profit. To make a profit of 10%, the selling price must be 110% of the cost price. Selling Price (for 10% profit) = 1.10 × CP Selling Price (for 10% profit) = 1.10 × 25,000 Selling Price (for 10% profit) = 27,500 shs The shopkeeper must sell the article at 27,500 shs to get a profit of 10%. b) Find the sum of the first four terms of a geometric progression which has a first term of 1 and a common ratio of (1)/(4). Step 1: Identify the given values. First term (a) = 1 Common ratio (r) = (1)/(4) Number of terms (n) = 4 Step 2: Use the formula for the sum of the first n terms of a geometric progression. Since |r| < 1, the formula is S_n = (a(1-r^n))/(1-r). Step 3: Substitute the values and calculate. S_4 = (1 (1 - (1)/(4))^4)1 - (1)/(4) S_4 = (1 - 1)/(256)(3)/(4) S_4 = (256 - 1)/(256)(3)/(4) S_4 = (255)/(256)(3)/(4) S_4 = (255)/(256) × (4)/(3) S_4 = (255 × 4)/(256 × 3) S_4 = (85 × 1)/(64 × 1) (since 255/3 = 85 and 256/4 = 64) S_4 = (85)/(64) The sum of the first four terms is (85)/(64). 4. MATHEMATICS a) The sum of the first six terms of an A.P is 72 and the second term is seven times the fifth term. Find the first term and the common difference. Step 1: Formulate equations from the given information. Let the first term be a and the common difference be d. The sum of the first n terms of an A.P. is S_n = (n)/(2)(2a + (n-1)d). For n=6, S_6 = 72: 72 = (6)/(2)(2a + (6-1)d) 72 = 3(2a + 5d) 24 = 2a + 5d (Equation 1) The n-th term of an A.P. is a_n = a + (n-1)d. Second term (a_2) = a + d. Fifth term ($a