Every progression is a sequence but the converse, i.e., every sequence is also a progression need not necessar

NCERT Exemplar · Q31

Q. Every progression is a sequence but the converse, i.e., every sequence is also a progression need not necessarily be true.

Sikkim CbseShort· 1mImportance★★★★★est

96% · 109/114 Questions

🔒 Locked · start free trial →

You're viewing a preview — the full solution, concept, methods & PYQ mapping are locked.

Start your 14-day free trial to unlock the full solution →

Concept understanding — Geometric Progression Arithmetic Progression

Comparing Arithmetic and Geometric Progressions

When you look at a sequence of numbers, the very first question to ask is: how does one term turn into the next? Two patterns cover almost every sequence you'll meet in Class 11 — adding a fixed number each time, or multiplying by a fixed number each time.

Two different rules of motion

An Arithmetic Progression (AP) moves by addition: each term is the previous term plus a constant common difference $d$ . If the first term is $a$ , the sequence is $a,\ a+d,\ a+2d,\ \dots$ , so the growth is a straight line — linear.
A Geometric Progression (GP) moves by multiplication: each term is the previous term times a constant common ratio $r$ . If the first term is $a$ , the sequence is $a,\ ar,\ ar^2,\ \dots$ , so the growth curves sharply upward (or shrinks toward zero) — exponential, not linear.

Note

This page is not the place to re-derive the individual formulas from scratch — the full nth-term and sum derivations for AP live under the Arithmetic Progression concept, and the equivalent derivations for GP live under the Geometric Progression concept. This page exists purely to help you tell the two patterns apart and choose the right one quickly.

The fastest way to tell them apart

Take any three consecutive terms $t_1, t_2, t_3$ of the sequence you're given:

Test for AP: compute $t_2-t_1$ and $t_3-t_2$ . If they're equal, you have an AP, and that equal value is $d$ .
Test for GP: compute $\dfrac{t_2}{t_1}$ and $\dfrac{t_3}{t_2}$ . If they're equal, you have a GP, and that equal value is $r$ .

If neither test succeeds, the sequence is neither an AP nor a GP on its own — it may still be built from a combination of the two (an arithmetico-geometric sequence), which needs its own separate technique.

Side-by-side

Feature	Arithmetic Progression	Geometric Progression
Rule to get the next term	Add $d$	Multiply by $r$
nth term	$a+(n-1)d$	$a\,r^{\,n-1}$
Sum of $n$ terms	$\dfrac{n}{2}[2a+(n-1)d]$	$a\,\dfrac{r^n-1}{r-1}$ (for $r\neq1$ )
Shape of growth	Straight line	Curve (exponential)
Allowed values of $d$ / $r$	Any real number, including $0$	Any real number except $0$

Start your 14-day free trial to unlock the full solution →

A progression is a sequence with a specific pattern (rule), but a sequence can be any ordered list of numbers without requiring a pattern. The statement is true.

The distinction between these two terms lies in the presence of a governing rule. Understanding this difference is fundamental to recognizing when we can apply progression formulas and when we're simply dealing with an arbitrary list.

A sequence is the most general concept: any ordered arrangement of numbers. Think of it as a list where position matters. The terms can follow any pattern, or no pattern at all. For instance, $1, 4, 9, 2, 7, 11, 3, \ldots$ is perfectly valid as a sequence—there's no requirement that consecutive terms relate in a predictable way.

A progression, on the other hand, is a special type of sequence where terms follow a definite rule or pattern. The three classical progressions you encounter in exams are:

Type	Pattern	Example
Arithmetic Progression (AP)	Constant difference between consecutive terms	$2, 5, 8, 11, 14, \ldots$
Geometric Progression (GP)	Constant ratio between consecutive terms	$3, 6, 12, 24, 48, \ldots$
Harmonic Progression (HP)	Reciprocals form an AP	$\frac{1}{2}, \frac{1}{4}, \frac{1}{6}, \frac{1}{8}, \ldots$

Now let's verify both directions of the statement:

Every progression is a sequence: This is always true. Since a progression is defined as a sequence with a specific pattern, it automatically satisfies the definition of a sequence. An AP like $3, 7, 11, 15, \ldots$ is certainly an ordered list of numbers, so it's a sequence. …

Start your 14-day free trial to unlock the full solution →

←Previous questionQ30 — Two sequences cannot be in both A.P. and G.P. together.→Next questionQ32 — Any term of an A.P. (except first) is equal to half the sum of terms w…

Q.Every progression is a sequence but the converse, i.e., every sequence is also a progression need not necessarily be true.

🎓Unlock everything free for 14 days

Unlock everything free for 14 days