Exercise Set 8.3
1. Find the 12th term of a GP with common ratio 2, whose 8th term is 192.
2. Find the 10th and nth terms of the GP: 5, 25, 125, … .
3. A sequence is given by the recursive rule t1 = 2, tn+1 = 3tn – 2 for n ≥ 1. Which term of the sequence is 730?
4. Which term of the GP: 2, 6, 18, … is 4374? Write the explicit formula as well as the recursive formula for the nth term.
5. A ball is dropped from a height of 80 metres. After hitting the ground, it bounces back to 60% of the height from which it fell. It continues bouncing in this way — each time rising to 60% of the previous height.
(i) What height does the ball reach after the 5th bounce?
(ii) What is the total vertical distance the ball has travelled by the time it hits the ground for the 6th time?
6. Which term of the sequence 2, 2√2 , 4, … is 128?
7. Fig. 8.12 shows Stages 0 to 3 of the Sierpiński square carpet. Stage 0 of this fractal is a square sheet of paper. To construct Stage 1, each side of the square is trisected and the points of trisection of opposite sides are joined to obtain nine smaller squares. The centre square is then removed and the 8 smaller squares are retained, leaving a square hole in the centre. The same process is repeated on the eight smaller shaded squares to obtain Stage 2 and so on.
Look at Fig. 8.12 and try to answer the following questions.
(i) How many red squares are there in Stages 0 to 3?
(ii) Can you predict the number of red squares in Stages 4 and 5?
(iii) Can you fnd a rule for the number of red squares at the nth stage? Write the explicit formula as well as the recursive formula for the number of red squares at any stage.
(iv) Suppose the area of the square in Stage 0 is 1 square unit.
What is the area of the red region in Stages 1, 2 and 3?
What will be the area of the red region in Stages 4 and 5?
Find the explicit as well as the recursive formula for the area of the red region at the nth stage. What happens to this area as n, the number of stages, goes on increasing?
End of Chapter Excercise
1. Find the 31st term of an AP whose 11th term is 38 and 16th term is 73.
2. Determine the AP whose third term is 16 and whose 7th term exceeds the 5th term by 12.
3. How many three-digit numbers are divisible by 7?
(Hint: All three-digit numbers divisible by 7 form an AP. Find the smallest and largest such three-digit numbers.)
4. How many multiples of 4 lie between 10 and 250?
(Hint: All multiples of 4 form an AP. Find the smallest and largest multiples of 4 between 10 and 250.)
5. Find a GP for which the sum of the frst two terms is – 4 and the ffth term is 4 times the third term.
6. Find all possible ways of expressing 100 as the sum of consecutive natural numbers.
7. The number of bacteria in a certain culture doubles every hour. If there were 30 bacteria present in the culture originally, how many bacteria will be present at the end of the 2nd hour, 4th hour and nth hour?
8. The sum of the 4th and 8th terms of an AP is 24 and the sum of the 6th and 10th terms is 44. Find the frst three terms of the AP.
9. Find the smallest value of n such that the sum of the first n natural numbers is greater than 1,000.
10. Which term of the GP: 2, 8, 32, … is 131072? Write the explicit formula as well as the recursive formula for the nth term.
11. The sum of the frst three terms of a GP is 13/12 and their product is –1. Find the common ratio and the terms.
12. If the 4th, 10th and 16th terms of a GP are x, y and z respectively, prove that x, y, z are in GP.
13. The sum of the frst three terms of a geometric progression is 26, and the sum of their squares is 364. Find the terms of the GP.
