Q1.
Can you think of any other kinds of sequences? List out five different types of sequences and discuss their properties with your friends.
Answer
Sequences differ in what their terms are made of and in how one term leads to the next. Five clearly different types:
| Type | Example | Property |
|---|---|---|
| Decreasing sequence of unit fractions | 1, 1/2, 1/3, 1/4, … | every term is positive, each smaller than the last, and the terms creep towards 0 without ever reaching it |
| Sequence of negative integers | –1, –2, –3, –4, … | terms fall without limit; n is still a natural number even though tn is negative |
| Alternating sequence | 1, –1, 1, –1, … | neither increasing nor decreasing; only two values, repeating for ever |
| Repeated-doubling sequence | 3, 6, 12, 24, 48, … | each term is a fixed multiple of the previous one, so it grows very fast |
| Sequence with no simple rule | 2, 3, 5, 7, 11, 13, … | the primes: perfectly well defined, yet with no formula that predicts the next term |
Why it happens: the definition of a sequence asks for only one thing — an ordered list. It says nothing about the terms rising, or about a formula existing. So a sequence may increase, decrease, oscillate, repeat, or wander; and the notation tn works in every case, because n merely labels the position while tn can be any real number at all.
Try This: write down the sequence of the digits of 22/7 in decimal, or the sequence of Indian Test cricket scores in an innings. Both are honest sequences; neither has a formula.