Class 9 - Chapter 8: Predicting What Comes Next: Exploring Sequences and Progressions

(i) t_n = 3n − 4

Substituting n = 1, 2, 3, 4, 5:

n	tn = 3n − 4
1	3(1) − 4 = −1
2	3(2) − 4 = 2
3	3(3) − 4 = 5
4	3(4) − 4 = 8
5	3(5) − 4 = 11

First Five Terms: −1, 2, 5, 8, 11

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(ii) t_n = 2 − 5n

n	tn = 2 − 5n
1	2 − 5 = −3
2	2 − 10 = −8
3	2 − 15 = −13
4	2 − 20 = −18
5	2 − 25 = −23

First Five Terms: −3, −8, −13, −18, −23

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(iii) t_n = n² − 2n + 3

n	tn = n2 − 2n + 3
1	12 − 2(1) + 3 = 2
2	22 − 2(2) + 3 = 3
3	32 − 2(3) + 3 = 6
4	42 − 2(4) + 3 = 11
5	52 − 2(5) + 3 = 18

First Five Terms: 2, 3, 6, 11, 18

Given: t_n = 5n − 3

10^th Term:

t₁₀ = 5(10) − 3

= 50 − 3

= 47

Answer: t₁₀ = 47

15^th Term:

t₁₅ = 5(15) − 3

= 75 − 3

= 72

Answer: t₁₅ = 72

Given: t_n = 5n − 3

Checking 97

5n − 3 = 97

5n = 100

n = 20

Since n is a whole number, 97 is the 20^th term.

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Checking 172

5n − 3 = 172

5n = 175

n = 35

Since n is a whole number, 172 is the 35^th term.

Given: t_n = 5n − 3

5n − 3 = 607

5n = 610

n = 122

Answer: 607 is the 122^nd term of the sequence.

Given: t₁ = −5

t_n+1 = t_n + 3

First Five Terms

Term	Value
t1	−5
t2	−2
t3	1
t4	4
t5	7

First Five Terms: −5, −2, 1, 4, 7

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Is 52 a Term?

This is an arithmetic sequence with:

a = −5, d = 3

t_n = a + (n − 1)d

52 = −5 + (n − 1)3

57 = 3(n − 1)

19 = n − 1

n = 20

Answer: Yes, 52 is the 20^th term.

T₁ = 1, T₂ = 2, T₃ = 4

T_n = T_n−1 + T_n−2 + T_n−3 for n ≥ 4

Term	Calculation	Value
T4	4 + 2 + 1	7
T5	7 + 4 + 2	13
T6	13 + 7 + 4	24
T7	24 + 13 + 7	44
T8	44 + 24 + 13	81

Answers:

T₄ = 7

T₅ = 13

T₆ = 24

T₇ = 44

T₈ = 81