5. Given the following sequence, determine whether it is arithmetic or not. If it is an Arithmetic Progression, find its general term : – 5, 2, 9, 16, 23, 30, ..... (3 marks)
Sol. – 5, 2, 9, 16, 23, 30, .....
t1 = – 5, t2 = 2, t3 = 9, t4 = 16, t5 = 23, t6 = 30
t2 – t1 = 2 – (– 5) = 2 + 5 = 7
t3 – t2 = 9 – 2 = 7
t4 – t3 = 16 – 9 = 7
t5 – t4 = 23 – 16 = 7
t6 – t5 = 30 – 23 = 7
Here, The difference between two consecutive terms is 7 which is a constant.
∴ The sequence is an A.P.
with a = t1 = – 5.
Common difference (d) = 7
tn = a + (n – 1) d
tn = – 5 + (n – 1) 7
tn = – 5 + 7n – 7
tn = 7n – 12
∴ The general term of A.P. is 7n – 12.
1. Check for a Common Difference
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Calculate the difference between consecutive terms:
- 2 - (-5) = 7
- 9 - 2 = 7
- 16 - 9 = 7
- 23 - 16 = 7
- 30 - 23 = 7
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Since the difference between any two consecutive terms is constant (7), the sequence is an arithmetic progression.
2. Find the General Term
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Identify the first term (a): a = -5
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Identify the common difference (d): d = 7
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Use the formula for the nth term (a<sub>n</sub>): a<sub>n</sub> = a + (n - 1) * d
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Substitute the values of 'a' and 'd': a<sub>n</sub> = -5 + (n - 1) * 7
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Simplify: a<sub>n</sub> = -5 + 7n - 7 a<sub>n</sub> = 7n - 12
Therefore, the general term of the arithmetic progression is a<sub>n</sub> = 7n - 12
