Sequence

A Sequence is a succession of terms in such a way that they are related to one another according to a well defined rule. However, the rules may differ depending on the arrangement of such terms.

This means that the rule that works for a particular sequence may not necessarily work for another.

Note; The nth term of a sequence is denoted as T_n where n stands for number of terms.

Examples of sequence are;

6, 11, 16, 21, 26,…(they differ by + 5)

1, 3, 9, 27, 81,…(they differ by × 3)

1, ³/_2, 2, ⁵/₂, 3,…(they differ by + ½ )

Examples;

1. The nth term of a sequence is given by 3 × 2^{n -2}. Write down the first three terms of the sequence.

Solution

Step 1

Remember that above the nth term is denoted as T_n.

T_n = 3 × 2^{n -2}

T₁ = 3 × 2^{1 -2}

T₁ = 3 × 2^-1

T₁ = 3 × ½ (Any number raise to power of negative 1 must become inverse, it’s a law!)

T₁ = ³/₂

Step 2

T₂ = 3 × 2^{2 -2}

T₂ = 3 × 2⁰ (Note; Any number raise to power of zero must be 1)

T₂ = 3 × 1

T₂ = 3

Step 3

T₃ = 3 × 2^{3 -2}

T₃ = 3 × 2¹

T₃ = 3 × 2

T₃ = 6

Step 4

 The first three terms of the sequence are ^3/₂, 3, 6, …

 
2. If the nth term of a sequence is denoted by the formula: n(2^{n + 1}) – 3n, find the sum of the first four terms.

Solution

Step 1

T_n = n(2^{n + 1}) – 3n ----This the nth term formula

T₁ = 1(2^{1 + 1}) – 3(1)

T₁ = 1(2²) – 3

T₁ = 1(2 × 2) – 3

T₁ = 1(4) – 3

T₁ = 4 – 3

T₁ = 1

Step 2

T₂ = 2(2^{2 + 1}) – 3(2)

T₂ = 2(2³) – 6

T₂ = 2(2 × 2 × 2) – 6

T₂ = 2(8) – 6

T₂ = 16 – 6

T₂ = 10

Step 3

T₃ = 3(2^{3 + 1}) – 3(3)

T₃ = 3(2⁴) – 9

T₃ = 3(2 × 2 × 2 × 2) – 9

T₃ = 3(16) – 9

T₃ = 48 – 9

T₃ = 39

Step 4

T₄ = 4(2^{4 + 1}) – 3(4)

T₄ = 4(2⁵) – 12

T₄ = 4(2 × 2 × 2 × 2 × 2) – 12

T₄ = 4(32) – 12

T₄ = 128 – 12

T₄ = 116

 The sum of the first four terms will be; 1 + 10 + 39 + 116 = 166.

                                                                                                     

3. Given the nth term of a sequence as

     

Find the first four terms.

Solution

Step 1

 

T₁ = - ¼

Step 2

T₂ = 0

Step 3

T₃ = ¾

Step 4

  

T₄ = 2

 The first four terms of the sequence are; - ¼, 0, ¾, 2, …

Practice these questions below;

1. The nth term of a sequence is denoted by 3n(2ⁿ – 1). Find the sum of the first five terms.

2. A particular term of sequence is represented by the formula 3 × 2^{n + 2}. What is the sum of the 5^th and 6^th terms?

3. Find the difference between the 4^th and 11^th terms of the sequence whose nth term is

4. Given that the nth term of a sequence is denoted by the formula;

Write down the first six terms.

5. What is the product of the 9^th and 12^th terms of the sequence with the nth term equals 4n² – 2^{n – 1}.