Does the list ever stop?

The idea

An AP with only a finite number of terms is a finite AP and has a last term; an AP whose terms go on without end is an infinite AP and has no last term.

Two lists

You save ₹50 every month, for 10 months: 50, 100, 150, … Now think of counting by 5: 5, 10, 15, 20, …

Both are made by adding a fixed number. But do both come to an end?

Which list has a last term?

• The savings list
• The counting list
• Both of them

Press Next term. Does it stop?

Two machines. Keep pressing Next term on each. One of them will tell you something. (+5 terms is a shortcut.)

Finite and infinite

An AP with only a finite number of terms is a finite AP, and it has a last term. An AP whose terms go on without end is an infinite AP, and it has no last term.

We write the dots "…" at the end of an infinite AP: 5, 10, 15, … A finite AP ends at its last term: 50, 100, 150, …, 500.

Notes

A finite AP has a last term. An infinite AP goes on without end and has no last term.

Check yourself

The heights (in cm) of some students standing in a queue are 147, 148, 149, …, 157. What kind of AP is it?

The rungs of a ladder are 45, 43, 41, 39, 37, 35, 33, 31 cm long. What kind of AP is it?

Which of these APs has no last term?

• Finite. It has a last term, 157. — correct. Yes! The list stops at 157, so it has a last term.
• Infinite. The heights keep growing.. An infinite AP has no last term. This list ends at 157.
• It is not an AP.. It is an AP: each term is 1 more than the one before (d = 1).
• Finite. It has a last term, 31. — correct. Yes! A ladder has a bottom and a top, so the list of rungs ends at 31.
• Infinite. The rungs keep getting shorter.. The list stops at 31, the top rung. It has a last term.
• The counting numbers 1, 2, 3, 4, … — correct. Yes! Counting never ends, so there is no last term.
• Prizes for classes I to XII: 200, 250, …, 750. It ends at 750, the prize for class XII. It has a last term.
• The savings after each of 10 months: 50, 100, …, 500. It ends at 500. It has a last term.