1. Problem: Identify the pattern of the sequence 2, 5, 10, 17, 26, ... and explain the reasoning. 2. The sequence is given as: 2, 5, 10, 17, 26, ... 3. To find the pattern, observe the differences between consecutive terms: $5 - 2 = 3$ $10 - 5 = 5$ $17 - 10 = 7$ $26 - 17 = 9$ 4. The differences are increasing by 2 each time: 3, 5, 7, 9, ... which suggests a quadratic pattern. 5. The general form for a quadratic sequence is: $$U_n = an^2 + bn + c$$ 6. Using the first three terms to find $a$, $b$, and $c$: For $n=1$, $U_1 = a(1)^2 + b(1) + c = a + b + c = 2$ For $n=2$, $U_2 = 4a + 2b + c = 5$ For $n=3$, $U_3 = 9a + 3b + c = 10$ 7. Solve the system: From $n=1$: $a + b + c = 2$ From $n=2$: $4a + 2b + c = 5$ From $n=3$: $9a + 3b + c = 10$ 8. Subtract first from second: $(4a + 2b + c) - (a + b + c) = 5 - 2$ $3a + b = 3$ 9. Subtract second from third: $(9a + 3b + c) - (4a + 2b + c) = 10 - 5$ $5a + b = 5$ 10. Subtract the two new equations: $(5a + b) - (3a + b) = 5 - 3$ $2a = 2 ightarrow a = 1$ 11. Substitute $a=1$ into $3a + b = 3$: $3(1) + b = 3 ightarrow b = 0$ 12. Substitute $a=1$, $b=0$ into $a + b + c = 2$: $1 + 0 + c = 2 ightarrow c = 1$ 13. Therefore, the formula is: $$U_n = n^2 + 1$$ 14. This matches the given answer $n^2 + 1$. Final answer: The pattern is $U_n = n^2 + 1$.