The start of a quadratic sequence is shown below. By first working out the \( n^{\text {th }} \) term rule, calculate the \( 10^{\text {th }} \) term of the sequence.

It appears that the sequence you're referring to wasn't included in your message. To determine the \( n^{\text{th}} \) term of a quadratic sequence and subsequently calculate the \( 10^{\text{th}} \) term, I'll need the initial terms of the sequence. **Here's how you can proceed once you have the sequence:** 1. **Identify the First Few Terms:** Let's assume the sequence starts with the terms \( a_1, a_2, a_3, \ldots \). 2. **Calculate the First Differences:** \[ d_1 = a_2 - a_1 \] \[ d_2 = a_3 - a_2 \] \[ \vdots \] 3. **Calculate the Second Differences:** \[ \Delta_1 = d_2 - d_1 \] \[ \Delta_2 = d_3 - d_2 \] \[ \vdots \] For a quadratic sequence, the second differences (\( \Delta \)) should be constant. 4. **Formulate the \( n^{\text{th}} \) Term:** The general form of a quadratic sequence is: \[ a_n = an^2 + bn + c \] Using the first few terms, set up equations to solve for constants \( a \), \( b \), and \( c \). 5. **Solve for \( a \), \( b \), and \( c \):** Substitute the values from the initial terms into the general formula to create a system of equations and solve for the constants. 6. **Calculate the \( 10^{\text{th}} \) Term:** Once you've determined \( a \), \( b \), and \( c \), plug \( n = 10 \) into the formula to find the \( 10^{\text{th}} \) term. **Example:** Suppose the sequence is: 3, 7, 13, 21, ... 1. **First Differences:** \[ 7 - 3 = 4 \] \[ 13 - 7 = 6 \] \[ 21 - 13 = 8 \] 2. **Second Differences:** \[ 6 - 4 = 2 \] \[ 8 - 6 = 2 \] (Constant second difference of 2) 3. **Form Equations:** \[ a(1)^2 + b(1) + c = 3 \quad \Rightarrow \quad a + b + c = 3 \] \[ a(2)^2 + b(2) + c = 7 \quad \Rightarrow \quad 4a + 2b + c = 7 \] \[ a(3)^2 + b(3) + c = 13 \quad \Rightarrow \quad 9a + 3b + c = 13 \] 4. **Solve the System:** Subtract the first equation from the second: \[ 3a + b = 4 \] Subtract the second equation from the third: \[ 5a + b = 6 \] Subtract these two new equations: \[ 2a = 2 \quad \Rightarrow \quad a = 1 \] Then: \[ 3(1) + b = 4 \quad \Rightarrow \quad b = 1 \] Finally: \[ 1 + 1 + c = 3 \quad \Rightarrow \quad c = 1 \] 5. **Final Formula:** \[ a_n = n^2 + n + 1 \] 6. **Calculate the \( 10^{\text{th}} \) Term:** \[ a_{10} = 10^2 + 10 + 1 = 100 + 10 + 1 = 111 \] **Please provide the initial terms of your sequence**, and I'd be happy to help you work out the \( n^{\text{th}} \) term and find the \( 10^{\text{th}} \) term specifically.