Question:

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? [Page No. 181]

Answer: Verified

A sequence is an arithmetic progression (AP) only if the difference between consecutive terms is constant. Checking each sequence:

Sequence	AP?	Reason
$1, 2, 3, 4, \dots$ (natural numbers)	Yes	constant difference $1$
$1, 3, 5, 7, \dots$ (odd numbers)	Yes	constant difference $2$
$1, 4, 7, 10, 13, \dots$	Yes	constant difference $3$
$-7, -3, 1, 5, 9, \dots$	Yes	constant difference $4$
$11, 7, 3, -1, -5, \dots$	Yes	constant difference $-4$
$1, 5, 9, 13, \dots$	Yes	constant difference $4$
$1, 3, 6, 10, 15, \dots$ (triangular)	No	differences $2, 3, 4, \dots$ vary
$1, 4, 9, 16, \dots$ (square)	No	differences $3, 5, 7, \dots$ vary
$1, \frac{1}{2}, \frac{1}{3}, \frac{1}{4}, \dots$ (unit fractions)	No	differences are not equal

In short, the sequences with a fixed common difference are APs, while those whose differences change (or which grow by multiplication, like a GP) are not.

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? [Page No. 181]

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