Q.Arithmetic and Geometric means of two positive numbers are 20 and 16 respectively. Find the numbers.

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.

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