Finding the Rule of a Quadratic Pattern

Finding the Rule of a Quadratic Pattern

For a quadratic rule $T_n=an^2+bn+c$, the constant second difference is $2a$. Find $a$ first, then substitute two known terms to solve for $b$ and $c$. Verify the completed rule with a third term. This turns pattern work into a short system of equations rather than trial and error.

Worked reasoning

Second difference 4 gives $2a=4$, so $a=2$. If $T_1=5$ and $T_2=12$, then $2+b+c=5$ and $8+2b+c=12$, giving $b=1,c=2$. Rule: $2n^2+n+2$.

Exam method

1. Use second difference to find a. 2. Substitute two term positions. 3. Solve b and c, then test a spare term.

Verify the general term

After finding a proposed $T_n$, test it with a term that was not used to determine the constants. If it works at positions 1, 2 and 5, confidence is much stronger than if it was only fitted to two values. Keep n as the term number, not the term value. In particular, substituting $n=1$ must produce the first term; that simple check catches many rules with a correct leading coefficient but an incorrect constant.

One-minute retrieval: Finding the Rule of a Quadratic Pattern

Close the worked solution. From memory, state its central rule, name one condition that makes it valid, and reconstruct one check that would catch a typical exam error.