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Example 

The second term of an arithmetic sequence is 1 and the seventh term is 26.

a Find the rst term and the common difference.

b Find the 1 00th term.

Answers

a u

2

= u

1

+ d = 1 u

7

= u

1

+ 6d = 26

u

7

 u

2

= 6d  d = 26  1 5d = 25

d = 5 u

1

+ d = 1

u

1

+ 5 = 1 u

1

= 4

The rst term is 4 and the common difference is 5.

b u

1 00

= u

1

+ 99d

= 4 + 99  5

= 491

Here you have two simultaneous equations. Solve them using algebra or a GDC.

Use the formula for the nth term with n = 100, u

1

= 4, d = 5.

Example 3

Here is a sequence of numbers 6 1 0 1 4 ... 50 a Write down the common difference.

b Find the number of terms in the sequence.

Answers a d = 4

b u

n

= 50  u

1

+ (n  1 )4 = 50 6 + (n  1 )4 = 50 (n  1 )4 = 44 (n  1 ) = 1 1 n = 1 2 So the sequence has 1 2 terms.

Use the formula for the nth term with u

1

= 6, d = 4. Solve for n.

Exercise 7A

EXAM -STY LE QU ESTIO N S

1 The rst four terms of an arithmetic sequence are 3 7 1 1 1 5

a Write down the eighth term in the sequence.

b Find the 1 50th term.

2 The third term of an arithmetic sequence is 8 and the ninth term is 26.

a Write down two equations in u

1

and d to show this information.

b Find the values of u

1

and d.

For solving simultaneous equations on a GDC, see Chapter 12, Section 1. 1.

GDC help on CD: Alternative dem o n stratio n s fo r th e TI-84 Plus an d Casio FX-9860GII GDCs are o n th e CD.

50 is the last term = u

n

Number and algebra 2

298

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EXAM -STY LE Q U ESTION S

3 The rst term of an arithmetic sequence is 12 and the ninth term is 16.

Calculate the value of the common difference.

4 The rst four terms of an arithmetic sequence are 3, 7, 1 1 , 1 5, 

a Write down the nth term of this sequence.

b Calculate the 50th term of this sequence.

5 The nth term of an arithmetic sequence is u

n

= 42  3n.

a Calculate the values of the rst two terms of this sequence.

b Which term of the sequence is 9 ?

c Two consecutive terms of this sequence, u

k

and u

k + 1

, have a sum of 33. Find k.

6 The sixth term of an arithmetic sequence is 34. The common difference is 6.

a Calculate the rst term of the sequence.

The nth term is 31 6.

b Calculate the value of n.

7 The rst term of an arithmetic sequence is 8 and the common difference is 7. The nth term is 393. Find the value of n.

8 Here is a nite sequence.

5 1 3 7 1 1  75

a Write down the value of the common difference.

b Find the 1 3th term.

c Find the number of terms in the sequence.

9 Here is a nite sequence.

8 1 0.5 1 3 1 5.5  1 88

a Write down the value of the common difference.

b Find the 1 2th term.

c Find the number of terms that the sequence has.

10 The nth term of a sequence is given by the formula u

n

= 1 2 + 7d.

a Write down the rst two terms.

b Write down the common difference.

c Find the 25th term.

The sum of the rst n terms of an arithmetic sequence

The sum of the rst n terms of an arithmetic sequence is called an

arithmetic series and is written as S

n

. S

n

= u

1

+ u

2

+ u

3

+ u

4

+ ... + u

n

Consecutive means the two terms are next to one another.

Carl Friedrich Gauss (17771855) is oten said to have been the greatest mathematician o the 19th century. Find out how Gauss worked out the sum o the frst 100 integers.

Chapter 7 299

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