3, 5, 7, 9, 11, ..... (1 mark)

(ii) 3, 5, 7, 9, 11, ..... (1 mark)

Sol. t₁ = 3, t2 = 5, t3 = 7, t4 = 9, t5 = 11

t₂ – t₁ = 5 – 3 = 2

t₃ – t₂ = 7 – 5 = 2

t₄ – t₃ = 9 – 7 = 2

t₅ – t₄ = 11 – 9 = 2

Here, The difference between ANY two consecutive terms is 2 which is constant.

∴ The sequence is an A.P.

1, 3, 6, 10, ..... (1 mark)

(i) 1, 3, 6, 10, ..... (1 mark)

Sol. t₁ = 1, t₂ = 3, t₃ = 6, t4 = 10

t₂ – t₁ = 3 – 1 = 2

t₃ – t₂ = 6 – 3 = 3

t₄ – t₃ = 10 – 6 = 4

Here, The difference between ANY two consecutive terms is not constant.

∴  The sequence is not an A.P.

(ix) 2, 4, 8, 16, ..... (1 mark)

(ix) 2, 4, 8, 16, ..... (1 mark)

Sol. t₁ = 2¹ = 2

t₂ = 2² = 4

t₃ = 2³ = 8

t₄ = 2⁴ = 16

t₅ = 2⁵ = 32

t₆ = 2⁶ = 64

t₇ = 2⁷ = 128

t₈ = 2⁸ = 256

∴  The next four terms of the sequence are 32, 64, 128 and 256.

(viii) – 25, – 23, – 21, – 19, ..... (1 mark)

(viii) – 25, – 23, – 21, – 19, ..... (1 mark)

Sol. t₁ = – 25

t₂ = – 25 + 2= – 23

t₃ = – 23 + 2= – 21

t₄ = – 21 + 2= – 19

t₅ = – 19 + 2= – 17

t₆ = – 17 + 2= – 15

t₇ = – 15 + 2= – 13

t₈ = – 13 + 2= – 11

∴ The next four terms of the sequence are – 17, – 15, – 13 and – 11.

Find the next four terms. 3, 9, 27, 81, ..... (1 mark)

Geometric Sequence Problem & Solution

(ii) Determine the next four terms of the sequence: \(3, 9, 27, 81, \dots\) (1 mark)

Solution:

Let the terms of the sequence be denoted by \(t_n\).

\(t_1 = 3^1 = 3\)

\(t_2 = 3^2 = 9\)

\(t_3 = 3^3 = 27\)

\(t_4 = 3^4 = 81\)

The general term of this geometric sequence is \(t_n = 3^n\). We need to find the next four terms, which are \(t_5, t_6, t_7, \text{ and } t_8\).

\(t_5 = 3^5 = 243\)

\(t_6 = 3^6 = 729\)

\(t_7 = 3^7 = 2187\)

\(t_8 = 3^8 = 6561\)

\(\therefore\) The next four terms of the sequence are 243, 729, 2187, and 6561.

For each sequence, find the next four terms. (i) 1, 2, 4, 7, 11, ..... (1 mark)

1. For each sequence, find the next four terms.

(i) 1, 2, 4, 7, 11, ..... (1 mark)

Sol. t₁ = 1 + 0 = 1

t₂ = 1 + 1 = 2

t₃ = 2 + 2 = 4

t₄ = 4 + 3 = 7

t₅ = 7 + 4 = 11

t₆ = 11 + 5 = 16

t₇ = 16 + 6 = 22

t₈ = 22 + 7 = 29

t₉ = 29 + 8 = 37

∴ The next four terms of the sequence are 16, 22, 29 and 37.