Consider all the sequences we have discussed so far. Which ones are arithmetic progressions and which ones are not? Can you justify your claim?

Step-by-step solution

Idea: A sequence is an AP when the difference between every pair of consecutive terms is the same. To show a sequence is not an AP, one pair of unequal differences is enough.

1. Test: work out (term − previous term) for several pairs. Same every time ⇒ AP; changes ⇒ not an AP.½ mark
2. APs: natural numbers 1, 2, 3, … (difference 1); odd numbers 1, 3, 5, … (difference 2; this is also 2n − 1); 1, 4, 7, 10, 13, … (difference 3).1 mark
3. Also APs: −7, −3, 1, 5, 9, … (difference 4); 5n − 2: 3, 8, 13, … (difference 5); 3n − 7: −4, −1, 2, … (difference 3); the squares pattern 1, 5, 9, 13, … (difference 4). From the exercises: 3n − 4: −1, 2, 5, … (difference 3); 2 − 5n: −3, −8, −13, … (difference −5); 5n − 3: 2, 7, 12, … (difference 5); −5, −2, 1, 4, … (difference 3). Any rule of the form (number) × n + (number) gives an AP.1 mark
4. Not APs: triangular numbers 1, 3, 6, 10 (differences 2, 3, 4); square numbers 1, 4, 9, 16 (differences 3, 5, 7); 6, 12, 24, 48, 96 (differences 6, 12, 24); 1, 12, 13, 14 (differences −12, −16, …).1 mark
5. Also not APs: the primes 2, 3, 5, 7 (differences 1, 2, 2); 1, 5, 13, 29 (differences 4, 8, 16); 3, 6, 30, 870; the Virahānka–Fibonacci sequence 1, 2, 3, 5, 8 (differences 1, 1, 2, 3); n² − 2n + 3: 2, 3, 6, 11 (differences 1, 3, 5); the running sums 1, 5, 12, 22 (differences 4, 7, 10); and 1, 2, 4, 7, 13 (differences 1, 2, 3, 6).½ mark

Answer to write in the exam

Common mistakes that cost marks

• Calling a sequence an AP because it increases. Square numbers increase but their differences (3, 5, 7, …) are not constant.
• Checking only the first difference. 1, 2, 4, 7, … starts with difference 1 but the next is 2.
• Thinking a decreasing sequence cannot be an AP. 11, 7, 3, −1, … is an AP with difference −4.

How this can come in the exam

Try one yourself

Which of these are APs? (a) 4, 9, 14, 19, … (b) 3, 9, 27, 81, … (c) 0, −2, −4, −6, …

More questions like this

• Verify that the following sequences are arithmetic progressions and write their nth terms. What do you observe when you plot the ordered pairs emerging from them?
• Using the formula t_n = a + (n − 1) × d, find the nth term of the following arithmetic progressions.
• Consider once again the AP: 1, 5, 9, 13, 17, … Since every term is 4 more than the previous term, another way of writing this sequence is t₁ = 1, t_n = t_n−1 + 4 for n ≥ 2. Verify for yourself that this recursive rule leads to the terms of the sequence.
• Find recursive rules for the APs in the previous exercises.
• A person books a taxi to travel in the city. The taxi company charges a fixed booking fee of ₹200 plus ₹40 per kilometre travelled. Let us write the sequence representing the total fare after travelling 1 km, 2 km, 3 km, and so on. If the person travels 10 km, what will be the total fare?

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