Chapter 5: Arithmetic Progressions

In a flower bed, there are 23 rose plants in the first row, 21 in the second, 19 in the third, and so on. There are 5 rose plants in the last row.

1. How many rows are there in the flower bed?
   (A) 10

   (B) 12

   (C) 11

   (D) 9
   Show Solution
   Solution: (A) 10
   Step 1: AP is 23, 21, 19, ..., 5. Here $a=23, d=-2, a_n=5$.
   Step 2: $5 = 23 + (n-1)(-2) \Rightarrow -18 = -2(n-1) \Rightarrow 9 = n-1 \Rightarrow n=10$.
2. How many rose plants are there in the 6th row?
   (A) 11

   (B) 13

   (C) 15

   (D) 17
   Show Solution
   Solution: (B) 13
   Step 1: $a_6 = a + 5d = 23 + 5(-2) = 23 - 10 = 13$.

Your friend wants to buy a mobile phone for ₹10,000. He saves ₹200 in the first week, ₹250 in the second week, ₹300 in the third week, and so on.

1. How much did he save in the 10th week?
   (A) ₹600

   (B) ₹650

   (C) ₹700

   (D) ₹750
   Show Solution
   Solution: (B) ₹650
   Step 1: AP is 200, 250, 300... Here $a=200, d=50$.
   Step 2: $a_{10} = 200 + 9(50) = 200 + 450 = 650$.
2. What is the total amount saved in 10 weeks?
   (A) ₹4000

   (B) ₹4250

   (C) ₹4500

   (D) ₹5000
   Show Solution
   Solution: (B) ₹4250
   Step 1: $S_{10} = \frac{10}{2}[2(200) + 9(50)] = 5[400 + 450] = 5(850) = 4250$.

A ladder has rungs 25 cm apart. The rungs decrease uniformly in length from 45 cm at the bottom to 25 cm at the top. The top and the bottom rungs are $2\frac{1}{2}$ m apart.

1. How many rungs are there?
   (A) 10

   (B) 11

   (C) 12

   (D) 13
   Show Solution
   Solution: (B) 11
   Step 1: Total distance = 2.5 m = 250 cm. Gap = 25 cm.
   Step 2: Number of rungs = $\frac{250}{25} + 1 = 10 + 1 = 11$.
2. What is the length of the wood required for the rungs?
   (A) 350 cm

   (B) 375 cm

   (C) 385 cm

   (D) 400 cm
   Show Solution
   Solution: (C) 385 cm
   Step 1: $a=45, l=25, n=11$.
   Step 2: Sum $= \frac{11}{2}(45+25) = \frac{11}{2}(70) = 11 \times 35 = 385$.