Sequences and series

Find the

n

th term and partial sums of arithmetic and geometric sequences; apply the limiting sum of an infinite geometric series; model growth, decay and financial situations using recurrence relations and series.

Algebra / Functions · MAV-12-03

Sub-topics

Arithmetic sequences and series

Find the

n

th term using

T_n = a + (n-1)d

and partial sums using

S_n = \frac{n}{2}(2a + (n-1)d)

; identify and work with arithmetic sequences in applied contexts.

Geometric sequences and series

Find the

n

th term using

T_n = ar^{n-1}

and partial sums using

S_n = \frac{a(r^n-1)}{r-1}

; evaluate the limiting sum

S_\infty = \frac{a}{1-r}

when

|r| < 1

Applications of series

Model growth, decay and financial problems using recurrence relations and geometric series; find limiting values of recurrence relations; apply compound interest and depreciation as geometric sequences.

Open Math

Sequences and series

Algebra / Functions · MAV-12-03

Name:

Date:

Arithmetic sequences and series

Q1Straightforward

An arithmetic sequence has first term

a = 4

and common difference

d = 7

. Find the 6th term

T_6

Q2Straightforward

The sequence

3, 7, 11, 15, \ldots

is arithmetic. What is the common difference

d

Q3Straightforward

The arithmetic sequence

10, 3, -4, -11, \ldots

has first term

a = 10

and common difference

d = -7

. Find

T_4

Q4Straightforward

Find the sum of the first 5 terms of the arithmetic series

2 + 5 + 8 + 11 + 14

Q5Moderate

An arithmetic sequence has first term

a = 7

and common difference

d = 4

. Which term equals 63? Find the value of

n

Q6Moderate

The 3rd term of an arithmetic sequence is 11 and the 8th term is 31. Find the common difference

d

Q7Moderate

The 3rd term of an arithmetic sequence is 11 and the common difference is 4. Find the first term

a

Q8Moderate

An arithmetic series has first term

a = 5

and common difference

d = 3

. Find

S_{10}

Q9Moderate

Find the sum of the arithmetic series

3 + 8 + 13 + \ldots + 123

Q10Moderate

A worker earns $1000 in their first week and receives a $50 pay rise each week. What are their total earnings over 8 weeks?

Q11Challenging

How many terms of the series

1 + 4 + 7 + \ldots

must be added for the sum to first exceed 100?

Q12Challenging

Three consecutive terms of an arithmetic sequence have a sum of 39 and a sum of squares of 509. Find the largest of the three terms.

Q13Challenging

The sum of the first

n

terms of a series is given by

S_n = 3n^2 + 2n

. Find the 7th term

T_7

Geometric sequences and series

Q14Straightforward

A geometric sequence has first term

a = 3

and common ratio

r = 2

. Find the 5th term

T_5

Q15Straightforward

The sequence

4, 12, 36, 108, \ldots

is geometric. What is the common ratio

r

Q16Straightforward

The sequence

256, 64, 16, 4, \ldots

is geometric. Find the 5th term

T_5

Q17Straightforward

Which of the following sequences is geometric?

A. $1, 3, 5, 7$ (each term increases by 2)
B. $2, 6, 18, 54$ (each term is multiplied by 3)
C. $5, 10, 15, 20$ (each term increases by 5)
D. $1, 2, 4, 7$ (differences are 1, 2, 3)

Q18Moderate

The 2nd term of a geometric sequence is 6 and the 5th term is 48. Find the common ratio

r

Q19Moderate

Find the sum of the first 6 terms of the geometric series

2 + 6 + 18 + \ldots

Q20Moderate

Find the limiting sum

S_\infty

of the geometric series

3 + 1 + \dfrac{1}{3} + \ldots

Q21Moderate

The limiting sum of an infinite geometric series is 20 and the first term is 5. Find the common ratio

r

. Express your answer as a fraction.

Q22Moderate

Find the sum of the 5-term geometric series

16 - 8 + 4 - 2 + 1

Q23Moderate

How many terms are in the geometric series

2 + 6 + 18 + \ldots + 4374

Q24Challenging

A ball is dropped from a height of 10 m. After each bounce it rises to 60% of its previous height. Find the total distance travelled by the ball before it comes to rest.

Q25Challenging

A geometric series has first term 8 and limiting sum 24. Find the 10th term, correct to 3 decimal places.

Q26Challenging

The infinite geometric series

x + x^2 + x^3 + \ldots

has a limiting sum when

|x| < 1

. Find the limiting sum when

x = 0.4

. Express your answer as a fraction in lowest terms.

Applications of series

Q27Straightforward

A sequence is defined by

T_1 = 5

and

T_n = 2T_{n-1}

. Find

T_4

Q28Straightforward

A sequence is defined by

T_1 = 200

and

T_n = T_{n-1} + 50

. Find

T_6

Q29Straightforward

$2000 is invested at

5\%

per annum compound interest. Find the value of the investment after 3 years. Give your answer to the nearest cent.

Q30Straightforward

A car is purchased for $20\,000 and depreciates at

12\%

per year. Find its value after 4 years, to the nearest dollar.

Q31Moderate

A salary starts at $60\,000 and increases by

4\%

each year. Find the total salary earned over 5 years, to the nearest dollar.

Q32Moderate

A machine's value follows the recurrence relation

V_n = 0.85\,V_{n-1}

with

V_0 = \$15\,000

. Find

V_3

to the nearest dollar.

Q33Moderate

The recurrence relation

T_n = 0.7\,T_{n-1} + 600

has a limiting value

L

n \to \infty

. Find

L

Q34Moderate

A population of 50\,000 grows at

3\%

per year. What will the population be after 10 years? Round to the nearest whole number.

Q35Moderate

$5000 is invested at

6\%

per annum compound interest. In which year does the investment first exceed $8000? (Find the smallest whole number

n

Q36Challenging

How many years does it take for an investment to double at

7\%

per annum compound interest? Give your answer to 2 decimal places.

Q37Challenging

A machine is purchased for $50\,000 and depreciates at

25\%

per year. After how many complete years is the machine worth less than $10\,000?

Q38Challenging

A superannuation fund receives a contribution of $5000 at the start of each year for 4 years, with interest compounding at

6\%

per annum. Find the total value of the fund at the end of 4 years, to the nearest dollar.

Worked solutions and answers at openmath.au/year-12/advanced/sequences-and-series