Work out the n-th term for the sequence: -1, 2, 7, 14, 23,…

Steps to Solve

1. Identify the Sequence Differences

First, find the first differences between the consecutive terms:

• $2 - (-1) = 3$
• $7 - 2 = 5$
• $14 - 7 = 7$
• $23 - 14 = 9$

The first differences are: $3, 5, 7, 9$.

2. Calculate the Second Differences

Next, calculate the second differences of the first differences:

• $5 - 3 = 2$
• $7 - 5 = 2$
• $9 - 7 = 2$

The second differences are constant and equal to $2$, which indicates that the sequence is quadratic.

3. Assume a Quadratic Formula

Assume the n-th term can be expressed in the form: $$ a_n = an^2 + bn + c $$

4. Set Up Equations

Use the known values to set up a system of equations. For the first few terms:

• When $n=1$: $a(1)^2 + b(1) + c = -1$ → $a + b + c = -1$ (Equation 1)
• When $n=2$: $a(2)^2 + b(2) + c = 2$ → $4a + 2b + c = 2$ (Equation 2)
• When $n=3$: $a(3)^2 + b(3) + c = 7$ → $9a + 3b + c = 7$ (Equation 3)

5. Solve the System of Equations

Now, solve Equations 1, 2, and 3:

Subtract Equation 1 from Equation 2: $$ (4a + 2b + c) - (a + b + c) = 2 - (-1) $$ $$ 3a + b = 3 \quad \text{(Equation 4)} $$

Subtract Equation 2 from Equation 3: $$ (9a + 3b + c) - (4a + 2b + c) = 7 - 2 $$ $$ 5a + b = 5 \quad \text{(Equation 5)} $$

Now subtract Equation 4 from Equation 5: $$ (5a + b) - (3a + b) = 5 - 3 $$ $$ 2a = 2 \Rightarrow a = 1 $$

Substituting $a = 1$ into Equation 4: $$ 3(1) + b = 3 \Rightarrow b = 0 $$

Substituting $a = 1$ and $b = 0$ into Equation 1: $$ 1 + 0 + c = -1 \Rightarrow c = -2 $$

6. Write the n-th Term Formula

The formula for the n-th term is: $$ a_n = n^2 - 2 $$