Q1.
Consider all the sequences we have discussed so far in this chapter. Which ones are arithmetic progressions and which ones are not? Can you justify your claim?
Answer
The test is single and mechanical: subtract each term from the one after it. If every one of those differences is the same number, the sequence is an AP; one exception is enough to rule it out.
| Sequence | Differences | AP? | Justification |
|---|---|---|---|
| 1, 2, 3, 4, 5, … | 1, 1, 1, … | Yes | a = 1, d = 1 |
| 1, 3, 5, 7, 9, … | 2, 2, 2, … | Yes | a = 1, d = 2 |
| 1, 4, 7, 10, 13, … | 3, 3, 3, … | Yes | a = 1, d = 3 |
| 1, 5, 9, 13, 17, … | 4, 4, 4, … | Yes | a = 1, d = 4 (Fig. 8.3) |
| 11, 7, 3, –1, –5, … | –4, –4, –4, … | Yes | a = 11, d = –4; d may be negative |
| –7, –3, 1, 5, 9, … | 4, 4, 4, … | Yes | a = –7, d = 4; terms may be negative |
| 1, 3, 6, 10, 15, … (triangular) | 2, 3, 4, 5 | No | the differences themselves change |
| 1, 4, 9, 16, 25, … (square) | 3, 5, 7, 9 | No | differences are the odd numbers, not a constant |
| 1, 1/2, 1/3, 1/4, … | –1/2, –1/6, –1/12 | No | the gaps shrink towards 0 |
| 2, 3, 5, 7, 11, 13, … (primes) | 1, 2, 2, 4, 2 | No | no regularity at all |
| 1, 2, 3, 5, 8, 13, … (Virahānka) | 1, 1, 2, 3, 5 | No | the differences are the sequence itself again |
| 1, 5, 13, 29, … (un = 2un–1 + 3) | 4, 8, 16 | No | differences double each time |
Why it happens: "increasing" and "arithmetic" are not the same thing. The triangular and square numbers rise steadily, yet neither is an AP, because being an AP is a statement about the gaps, not about the terms. Equivalently, an explicit rule of the form tn = an + b (linear in n) always gives an AP, and any rule with n2, or with n in an exponent, never does.
Tip: checking one pair of terms proves nothing. 1, 3, 6 begins with a difference of 2 but is not an AP — you must check that every difference matches.