Sequence Finding 0E4E85

1. **State the problem:** We are given the sequence $1$, $2$, $4$, $7$ and need to find the pattern or formula that generates this sequence. 2. **Analyze the sequence:** Look at the differences between consecutive terms: - $2 - 1 = 1$ - $4 - 2 = 2$ - $7 - 4 = 3$ 3. **Check the second differences:** - $2 - 1 = 1$ - $3 - 2 = 1$ Since the second differences are constant ($1$), this suggests the sequence can be described by a quadratic formula of the form: $$a_n = An^2 + Bn + C$$ 4. **Set up equations using the first three terms:** For $n=1$, $a_1 = A(1)^2 + B(1) + C = A + B + C = 1$ For $n=2$, $a_2 = 4A + 2B + C = 2$ For $n=3$, $a_3 = 9A + 3B + C = 4$ 5. **Solve the system:** From the first equation: $A + B + C = 1$ From the second: $4A + 2B + C = 2$ From the third: $9A + 3B + C = 4$ Subtract the first from the second: $$ (4A + 2B + C) - (A + B + C) = 2 - 1 \Rightarrow 3A + B = 1 $$ Subtract the second from the third: $$ (9A + 3B + C) - (4A + 2B + C) = 4 - 2 \Rightarrow 5A + B = 2 $$ Subtract the two new equations: $$ (5A + B) - (3A + B) = 2 - 1 \Rightarrow 2A = 1 \Rightarrow A = \frac{1}{2} $$ Substitute $A = \frac{1}{2}$ into $3A + B = 1$: $$ 3 \times \frac{1}{2} + B = 1 \Rightarrow \frac{3}{2} + B = 1 \Rightarrow B = 1 - \frac{3}{2} = -\frac{1}{2} $$ Substitute $A$ and $B$ into $A + B + C = 1$: $$ \frac{1}{2} - \frac{1}{2} + C = 1 \Rightarrow C = 1 $$ 6. **Write the formula:** $$ a_n = \frac{1}{2} n^2 - \frac{1}{2} n + 1 $$ 7. **Verify with $n=4$:** $$ a_4 = \frac{1}{2} \times 16 - \frac{1}{2} \times 4 + 1 = 8 - 2 + 1 = 7 $$ This matches the given term. **Final answer:** $$ a_n = \frac{1}{2} n^2 - \frac{1}{2} n + 1 $$