Patterns and conjectures

A conjecture is a guess based on evidence. Mathematicians often spot a pattern, guess what rule produces it, and then try to prove the guess. Sometimes a single counter-example destroys it; sometimes it survives every test and becomes a theorem.

Idea

Begin with small examples: $1 = 1$ $1 + 3 = 4$ $1 + 3 + 5 = 9$ $1 + 3 + 5 + 7 = 16$

The right-hand sides are $1, 4, 9, 16, \dots$ , the squares. Conjecture: the sum of the first $n$ odd numbers is $n^2$. Test with $n = 5$: $1+3+5+7+9 = 25 = 5^2$ ✓. Test with $n = 6$: $1+3+5+7+9+11 = 36 = 6^2$ ✓. The pattern keeps holding. A clean visual proof: arrange $n$ rows of dots; each row adds one more odd-number worth of dots, building a square.

Another classic: $1 = 1$ $1 + 2 = 3$ $1 + 2 + 3 = 6$ $1 + 2 + 3 + 4 = 10$ The RHS are $1, 3, 6, 10$ , triangular numbers. Conjecture: $1 + 2 + \dots + n = \tfrac{n(n+1)}{2}$. Check at $n = 5$: $\tfrac{5 \times 6}{2} = 15 = 1+2+3+4+5$ ✓.

Method for working with patterns:

1. Generate examples. Compute the first $4$–$6$ terms.
2. Spot regularity. Look at differences, ratios, parities.
3. Conjecture a formula. Test it on the next two terms.
4. Try to justify. Why does it work? A picture, a small proof, an algebraic argument.

Even if you cannot prove your conjecture, the practice of carefully spotting and testing is valuable in itself.

Worked examples

Example 1. Consider $1, 4, 9, 16, 25, \dots$. What is the $n$-th term?

These are $1^2, 2^2, 3^2, \dots$ , the squares. So $a_n = n^2$.

Example 2. Spot the pattern: $2, 5, 10, 17, 26, \dots$. Find the next two and a formula.

Differences are $3, 5, 7, 9, \dots$ , odd numbers, increasing by $2$. So next differences $11, 13$, giving next two terms $26+11=37$ and $37+13=50$. Formula guess: $a_n = n^2 + 1$. Check: $1+1=2, 4+1=5, 9+1=10$ ✓.

Example 3. Conjecture: "the sum of three consecutive numbers is always divisible by $3$." Justify.

Three consecutive: $n, n+1, n+2$. Sum $= 3n + 3 = 3(n+1)$. Always a multiple of $3$. ✓

Example 4. Consider triangular numbers $T_n = 1 + 2 + \dots + n$. Show $T_n + T_{n+1} = (n+1)^2$.

$T_n + T_{n+1} = \tfrac{n(n+1)}{2} + \tfrac{(n+1)(n+2)}{2} = \tfrac{(n+1)(n + n + 2)}{2} = \tfrac{(n+1)(2n+2)}{2} = (n+1)^2$. ✓

Try it yourself

1. Find the next two terms of $1, 8, 27, 64, \dots$. State the formula.
2. Pattern: $1, 3, 7, 15, 31, \dots$. Find $a_6$ and conjecture a formula.
3. Compute $1 + 3 + 5 + 7 + 9$. Is it a perfect square?
4. Sum of first $20$ odd numbers , without adding them, what is it?
5. Sum the first $10$ natural numbers. Use the formula.
6. Is the sum of two triangular numbers always a square? (Try.)
7. Conjecture: "the product of two consecutive numbers is always even." Justify.
8. Find the pattern: $3, 9, 27, 81, \dots$. What is the $7$th term?

Activity

Pick three favourite patterns from this chapter. For each, write the first $5$ terms, the rule, a small proof or justification. Make a "patterns poster" for your classroom.