HOW TO FIND THE POSITION OF A TERM IN AN ARITHMETIC SEQUENCE

Problem 1 :

1) For each question use the rule to find the first five terms of the sequence.

2) Can you find the 10^th term of the sequence ?

3) Can you find the 100^th term of the sequence ?

Rule :

2n - 1

Solution :

Given rule = 2n - 1

1^st term = 1, 2^nd term = 3, 3^rd term = 5, 4^th term = 7 and 5^th term = 9

10^th term :

= 2(10) - 1

= 20 - 1

= 19

100^th term :

= 2(100) - 1

= 200 - 1

= 199

Problem 2 :

Write the term to term rule for the sequence given below, then work out the next two terms.

4, 9, 14, 19, 24, .......

a) Is the number 37 in the sequence.

Solution :

4, 9, 14, 19, 24, .......

Every term is added with 5, by creating the rule

a_n = 5n - 1

Finding next two terms :

The next two terms are 24 and 29.

a) Is 34 in the sequence :

Since we don't know whether 34 is in the sequence or not, we can consider a_n = 34

34 = 5n - 1

5n = 34 + 1

5n = 35

n = 35/5

n = 7

From this, it is clear 7^th term of the sequence is 34.

Problem 3 :

Calculate the difference between the 10^th term and 50^th term of the sequence

9, 14, 19, 24, ... ...

Solution :

9, 14, 19, 24, ... ...

In every term 5 is added. Creating the general term, we get

a_n = 5n + 4

Problem 4 :

Find the n^th term for each of the following sequences

1/2, 3/4, 5/6, 7/8, .............

Solution :

By observing each values in the numerator and denominator, numerators are odd values and denominators are even numbers.

Odd values = 2n - 1

Even values = 2n

So, n^th term of the sequence is (2n - 1)/2n

Problem 5 :

Write the next three terms of the arithmetic sequence.

a) First term: 2 Common difference: 13

b) First term: 18 Common difference: −6

Solution :

a) a = 2, d = 13

2, 2 + 13, 15 + 13, ......

2, 15, 28, ...............

b) a = 18, d = -6

18, 18 - 6, 12 - 6,...........

18, 12, 6, .............

Problem 6 :

write an equation for the nth term of the arithmetic sequence. Then find a₁₀.

−5, −4, −3, −2, . . .

Solution :

a = -5, d = -4 - (-5)

= -4 + 5

d = 1

a₁₀ = a + 9d

= -5 + 9(1)

= -5 + 9

= 4

So, the 10th term is 4.

Problem 7 :

Describe and correct the error in finding the common difference of the arithmetic sequence.

Solution :

Common difference = 1 - 2 ==> -1

0 - 1 ==> -1

-1 - 0 ==> -1

Bu the given common difference is 1 and that is the error.

Problem 8 :

Describe and correct the error in writing an equation for the nth term of the arithmetic sequence.

Solution :

a = 14, d = 22 - 14

= 8

nth term :

an = a + (n- 1) d

= 14 + (n - 1)8

= 14 + 8n - 8

= 6 + 8n

By the given nth term is different and that is the error.

Problem 9 :

The first term of an arithmetic sequence is 3. The common difference of the sequence is 1.5 times the first term. Write the next three terms of the sequence. Then graph the sequence

Solution :

a = 3

d = 1.5(8)

= 12

3, 3 + 12, 15 + 12, ...........

3, 15, 27, ...............