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Question 4
Number Sequences [General Term 2ⁿ, 3ⁿ and 4ⁿ etc] Class-Activity

1. Study: In the sequence 2, 4, 8, 16;... there is not a constant difference between each term.
This is because each term is multiplied by 2 to get to the next term.
Hence the next two terms are: 2, 4, 8, 16, 32, 64

If we write 2, 4, 8, 16;... in expanded form we get:
2; 2x2; 2x2x2; 2x2x2x2;
or 2¹; 2²; 2³; 2⁴; ... In exponential form. The 4th term is 2⁴ thus the nth term is 2ⁿ.

2. Fill in the missing values in each table. use exponential form, where necessary.
a) position | 1 | 2 | 3 | 4 | n
---|---|---|---|---|---|
value | 3 | 9 | 27 | 81 | 3ⁿ

b) position | 1 | 2 | 3 | 4 | n
---|---|---|---|---|---|
value | 4 | 16 | 64 | 256 | 4x7 | 4x10

c) position | 1 | 2 | 3 | 4 | n
---|---|---|---|---|---|
value | 2 | 4 | 8 | 16 | 2x10 | 2x12

d) position | 1 | 2 | 3 | 4 | n
---|---|---|---|---|---|
value | 5 | 25 | 125 | 625 | 5x8 | 15x5

3. Determine the value of the 5th, nth and 10th term in each number sequence,
a) 2; 4; 8; 16; ...
5th term =
nth term = 2ⁿ
10th term = ...

b) 3; 9; 27; 81; ...
5th term =
nth term = 3xⁿ
10th term = ...

c) 6; 6²; 6³; 6⁴; ...
5th term =
nth term = 6xⁿ
10th term = ...

d) 9; 9²; 9³; 9⁴; ...
5th term =
nth term = 9xⁿ
10th term = ...

Answer :

In this question, we are examining number sequences and understanding how to find the nth term and specific terms in exponential sequences. Let's look at each part in detail:

  1. Filling in Missing Values:
    a) The sequence given is 3, 9, 27, 81, ... which can be written in exponential form as:

    • 3¹, 3², 3³, 3⁴, ...
    • So, the nth term is [tex]3^n[/tex].

    b) Looking at the pattern for the second table:

    • Start with 4: [tex]4 = 4^1[/tex]
    • Next, 16: [tex]16 = 4^2[/tex]
    • Then, 64: [tex]64 = 4^3[/tex]
    • Next, 256: [tex]256 = 4^4[/tex]
    • Therefore, the nth term in exponential form is [tex]4^n[/tex].

    c) For this sequence:

    • 2, 4, 8, 16, ... can be expressed as: [tex]2^1, 2^2, 2^3, 2^4, ...[/tex]
    • Hence, the nth term is [tex]2^n[/tex].

    d) Finally, the sequence:

    • 5, 25, 125, 625, ... corresponds to: [tex]5^1, 5^2, 5^3, 5^4, ...[/tex]
    • Thus, the nth term is [tex]5^n[/tex].

  2. Determining the 5th, nth, and 10th Term:

    a) For the sequence 2, 4, 8, 16, ...:

    • The 5th term is [tex]2^5 = 32[/tex]
    • The nth term is [tex]2^n[/tex]
    • The 10th term is [tex]2^{10} = 1024[/tex]

    b) For the sequence 3, 9, 27, 81, ...:

    • The 5th term is [tex]3^5 = 243[/tex]
    • The nth term is [tex]3^n[/tex]
    • The 10th term is [tex]3^{10} = 59049[/tex]

    c) For the sequence 6, 6², 6³, 6⁴, ...:

    • The 5th term is [tex]6^5 = 7776[/tex]
    • The nth term is [tex]6^n[/tex]
    • The 10th term is [tex]6^{10} = 60466176[/tex]

    d) For the sequence 9, 9², 9³, 9⁴, ...:

    • The 5th term is [tex]9^5 = 59049[/tex]
    • The nth term is [tex]9^n[/tex]
    • The 10th term is [tex]9^{10} = 3486784401[/tex]

When studying number sequences like these, notice the pattern based on powers of a number. Identifying the base and recognizing the pattern as an exponential sequence is essential to finding terms at specific positions efficiently.

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