Sequences and Series quiz

The common ratio of the GP 3, -6, 12, ... is:

The sum of the infinite GP 6 + 3 + 1.5 + ... is:

If the AM and GM of two positive numbers are 10 and 8, the numbers are:

The 5th term of the GP 1, 3, 9, ... is:

The sum to infinity of a GP exists when:

If three numbers in GP have product 216, the middle number is:

The GM of 2 and 8 is:

The sum 1 + 2 + 4 + ... + 2^9 is:

The nth term of a GP with first term a and ratio r is:

The 20th term of the GP 2, 1, 1/2, ... is:

Write the first five terms of the sequence a1 = 3, an = 3a(n-1) + 2. (2 marks)

Find the 12th term of the GP whose 8th term is 192 and common ratio is 2. (2 marks)

Find the sum to infinity of 1 - 1/3 + 1/9 - 1/27 + .... (2 marks)

Find the GM of 3 and 27. (2 marks)

Find the sum of the GP 1, -a, a^2, -a^3, ... to n terms, where a is not -1. (2 marks)

Explain why the series 1 + 2 + 4 + 8 + ... has no sum to infinity. (2 marks)

If a, b and c are in GP, show that b^2 = ac. (2 marks)

The first term of a GP is 1 and the sum of its third and fifth terms is 90. Find the common ratio. (2 marks)

Find the number of terms of the GP 3, 3/2, 3/4, ... needed to give a sum of 3069/512. (2 marks)

Write the general term of the sequence 2, 5, 8, 11, .... (2 marks)

Find the sum to n terms of 8 + 88 + 888 + .... (3 marks)

The sum of the first three terms of a GP is 39/10 and their product is 1. Find the terms. (3 marks)

Find the sum of the first 10 terms of the GP 1, 2, 4, 8, .... (3 marks)

How many terms of the GP 3, 3^2, 3^3, ... are needed to give the sum 120? (3 marks)

Rs 500 is invested at 10% compound interest per year. Find the amount after 10 years, taking 1.1^10 = 2.5937. (3 marks)

Read the passage and answer the questions. A culture has 30 bacteria at the start, and the number doubles every hour. (i) Find the number after 2 hours. (ii) Find the number after 4 hours. (iii) Write the number after n hours. (5 marks)

Read the passage and answer the questions. A ball is dropped from a height of 10 m. Each time it hits the ground it rebounds to 3/4 of the height from which it fell. (i) Find the height of the first rebound. (ii) Write the rebound heights as a GP. (iii) Find the total distance travelled before the ball comes to rest. (5 marks)

Read the passage and answer the questions. Asha saves Rs 200 in the first month and increases her saving by Rs 40 every month. (i) Find her saving in the 12th month. (ii) Find her total saving in 12 months. (iii) Is this an AP or a GP? (5 marks)

Read the passage and answer the questions. A sheet of paper 0.1 mm thick is folded in half again and again. (i) Write the thickness after each of the first three folds. (ii) Find the thickness after 10 folds. (iii) Identify the common ratio. (5 marks)

Read the passage and answer the questions. A town of 20,000 people grows at 5% per year. (i) Write the population after 1 and 2 years. (ii) Show that the yearly populations form a GP. (iii) Write the population after n years. (5 marks)

Define a geometric progression. Derive its general term and the sum of its first n terms. (6 marks)

Derive the sum to infinity of a GP when -1 < r < 1. Use it to express 0.333... as a fraction. (6 marks)

Explain how to insert n geometric means between two positive numbers a and b. Insert 4 GMs between 3 and 96. (6 marks)

Prove that AM >= GM for two positive numbers. When does equality hold? Show that if a and b are positive with a + b = 10, then ab <= 25. (6 marks)

Explain with examples how geometric progressions model compound interest and population growth. Why is the AP model unsuitable for these? (6 marks)