Every progression is a sequence but the converse, i.e., every sequence is also a progression need not necessarily be true.
Any term of an AP (except first) is equal to half the sum of terms which are equidistant from it.
The sum or difference of two GP, is again a GP.
If the sum of $n$ terms of a sequence is quadratic expression, then it always represents an AP.
Match the following.
| Column I | Column II | ||
|---|---|---|---|
| (i) | $4,1, \frac{1}{4}, \frac{1}{16}$ | (a) | AP |
| (ii) | $2,3,5,7$ | (b) | Sequence |
| (iii) | $13,8,3,-2,-7$ | (c) | GP |
(i) $4,1, \frac{1}{4}, \frac{1}{16}$
$\Rightarrow \quad \frac{T_2}{T_1}=\frac{1}{4}\Rightarrow \frac{T_3}{T_2}=\frac{1}{4}\Rightarrow \frac{T_4}{T_3}=\frac{1/16}{1/4}=\frac{1}{4}$
Hence, it is a GP.
(ii) $2,3,5,7$
$$\begin{array}{ll} \because & T_2-T_1=3-2=1 \\ & T_3-T_2=5-3=2 \\ \because & T_2-T_1 \neq T_3-T_2 \end{array}$$
Hence, it is not an AP.
$$\begin{array}{ll} \text { Again, } & \frac{T_2}{T_1}=3 / 2 \Rightarrow \frac{T_3}{T_2}=5 / 3 \\ \because & \frac{T_2}{T_1} \neq \frac{T_3}{T_2} \end{array}$$
It is not a GP.
Hence, it is a sequence.
(iii) $13,8,3,-2,-7$
$$\begin{aligned} & T_2-T_1=8-13=-5 \\ & T_3-T_2=3-8=-5 \\ \because \quad & T_2-T_1=T_3-T_2 \end{aligned}$$
Hence, it is an AP.