Question

A school is organizing a charity run to raise funds for a local hospital. The run is planned as a series of rounds around a track. with each round being 300 metres. To make the event more challenging and engaging. the organizers decide to increase the distance of each subsequent round by 50 metres. For example, the second round will be 350 metres, the third round will be 400 metres and so on. The total number of rounds planned is 10.

Based on the information given above, answer the following questions:

1. Write the fourth, fifth, and sixth terms of the Arithmetic Progression so formed.
2. Determine the distance of the 8^th round. (1)
3. 1. Find the total distance run after completing all 10 rounds. (2)
      OR
   2. If a runner completes only the first 6 rounds, what is the total distance run by the runner? (2)

Answer 1

We are given an arithmetic progression (AP) where:

The first term a = 300 meters

The common difference d = 50 meters

Total number of rounds n = 10

(i) The general formula for the n^th term of an AP is:

a_n = a + (n − 1)d

a₄ = 300 + (4 − 1)50

= 300 + 150

= 450 metres

a₅ = 300 + (5 − 1)50

= 300 + 200

= 500 metres

a₆ = 300 + (6 − 1)50

= 300 + 250

= 550 metres

(ii) a₈ = 300 + (8 − 1)50

= 300 + 350

= 650 metres

(iii) (a) The sum of an AP is given by:

S_n = `n/2 xx (2a + (n - 1)d)`

for n = 10

S₁₀ = `10/2 xx (2(300) + (10 - 1) xx 50)`

= 5 × (600 + 9 × 50)

= 5 × (600 + 450)

= 5 × 1050

= 5250 metres

(iii) (b) Using the same formula for S_n

S₆ = `6/2 xx (2(300) + (6 - 1) xx 50)`

= 3 × (600 + 5 × 50)

= 3 × (600 + 250)

= 3 × 850

= 2550 metres