Word problems on Arithmetic Progression

Question 21

Question:   There are 25 trees at equal distances of 5 metres in a line with a well, the distance of the well from the nearest tree being 10 metres. A gardener waters all trees separately starting from the well and he returns to the well after watering each tree to get water for the next. Find the total distance the gardener will cover in order to water all the trees.

Answer:   
Let O be the position of the well (assuming that the trees and the well are collinear). T₁, T₂, T₃......T₂₅ be the position of trees. The total distance covered by the gardener will be:

Question 22

Question:   A man saves Rs 32 during the first year, Rs 36 in the next year and Rs 40 in the third year. If he continues his savings in the sequence, in how many years will he save Rs 2000.

Answer:    Let a = 32, d = 4, S_n = 2000







Since the duration is positive, n = 25 years.

Question 23

Question:   An insect starts from a point and travels in a straight-line path one mm in the first second and half of the distance covered in the previous second in the succeeding second. In how much time would it reach a point 3mm from its starting point?

Answer:   
Let time taken to cover a distance of 3mm be n seconds. The series of the given distances covered from a G P. where




insect would never reach its goal.

Question 24

Question:   For each geometric progression, find the common ratio. Then tell whether the ratio is a growth factor or decay factor or neither. Find also the n^th term.
i. 5, 10, 20, 40,....
ii.6, 0.6, 0.06,....

Answer:    i. 5, 10, 20, 40,....


Since r > 1, r is a growth factor.
ii. 6, 0.6, 0.06,....


Since r = 0.1 < 1, r is a decay factor.

Question 25

Question:   Use the given values for a and r to find the first four terms of a GP and the n^th term.
i. a = 1234, r = 0.1


iv. a = 1, r = 1

Answer:   

Question 26

Question:   The sum of 3 numbers in GP is 35 and their product is 1000. Find the numbers.

Answer:   

Question 27

Question:   If a² + b², ab + bc and b² + c² are in G.P. Prove that a, b, c are also in G.P

Answer:    a² + b², ab + bc, b² + c²are in G.P

Question 28

Question:   If a, b, c, d are in GP, prove that

Answer:   

Question 29

Question:  
prove that x, y and z are in GP.

Answer:   

Question 30

Question:   If one Geometric mean G and two arithmetic means p and q be inserted between two quantities, show that G²=(2p -q)(2q - p).

Answer:    Let the two numbers be a and b
G is the GM between a and b
Then G²= ab ...(1)
p, q are the AM's between a and b.
Then a, p, q and b are in AP
2p = a+q and 2q = b+p