Arithmetic and Geometric Sequences: practice questions

6 questions · 12 marks · free, no account · Read the notes first

1. Question 1Multiple choice · 1 mark

   Find the 20th term of the arithmetic sequence $3, 7, 11, \ldots$

   A

   $83$

   B

   $79$

   C

   $80$

   D

   $76$

2. Question 2Multiple choice · 1 mark

   Find the sum of the first 6 terms of the geometric sequence $2, 6, 18, \ldots$

   A

   $486$

   B

   $1458$

   C

   $728$

   D

   $364$

3. Question 3Multiple choice · 1 mark

   Find the sum to infinity of $12, 6, 3, \ldots$

   A

   $24$

   B

   $21$

   C

   $36$

   D

   It does not exist

4. Question 4Multiple choice · 1 mark

   In an arithmetic sequence, the 3rd term is 10 and the 7th term is 22. Find the common difference.

   A

   $4$

   B

   $6$

   C

   $12$

   D

   $3$

5. Question 54 marks

   Find the least number of terms of $5 + 9 + 13 + \cdots$ whose sum exceeds 1000.

   Show worked solution

   This is an A.S. with $a = 5$ and $d = 4$.

   $$S(n) = \frac{n}{2}[2(5) + (n - 1)(4)] = n(2n + 3) \qquad \text{1M}$$ $$2n^2 + 3n > 1000 \qquad \text{1M}$$

   $n > 21.6$ (or $n < -23.1$, rejected) 1A

   The least number of terms is 22. 1A

6. Question 64 marks

   A ball is dropped from a height of 8 m. Each time it hits the ground it rebounds to $\frac{3}{4}$ of the height from which it fell. Find the total distance travelled by the ball before it comes to rest.

   Show worked solution

   The rebound heights $6, 4.5, 3.375, \ldots$ form a G.S. with $a = 6$ and $r = \frac{3}{4}$. 1M

   Sum of the rebound heights $= \dfrac{6}{1 - \frac{3}{4}} = 24$ m 1M

   The ball goes up and down each rebound height:

   Total distance $= 8 + 2(24)$ 1M

   $= 56$ m 1A