Let's start by determining the value of [tex]\( d \)[/tex] using the given general rule [tex]\( T_n = d n^2 + 1 \)[/tex] and the values in the table.
### Step 1: Determine [tex]\( d \)[/tex]
From the table:
- [tex]\( T_1 = 3 \)[/tex]
- [tex]\( T_2 = 9 \)[/tex]
- [tex]\( T_3 = 19 \)[/tex]
- [tex]\( T_4 = 33 \)[/tex]
Let's use the first term [tex]\( T_1 \)[/tex] to find [tex]\( d \)[/tex]:
[tex]\[ T_1 = d \cdot 1^2 + 1 = 3 \][/tex]
This simplifies to:
[tex]\[ d \cdot 1 + 1 = 3 \implies d + 1 = 3 \implies d = 2 \][/tex]
Thus, the value of [tex]\( d \)[/tex] is [tex]\( 2 \)[/tex].
### Step 2: Find the Values of [tex]\( A \)[/tex], [tex]\( B \)[/tex], and [tex]\( C \)[/tex]
Using [tex]\( d = 2 \)[/tex], we can now find the values for [tex]\( T_5 \)[/tex], [tex]\( T_{11} \)[/tex], and [tex]\( T_{20} \)[/tex].
For [tex]\( T_5 \)[/tex]:
[tex]\[ T_5 = 2 \cdot 5^2 + 1 = 2 \cdot 25 + 1 = 50 + 1 = 51 \][/tex]
So, [tex]\( A = 51 \)[/tex].
For [tex]\( T_{11} \)[/tex]:
[tex]\[ T_{11} = 2 \cdot 11^2 + 1 = 2 \cdot 121 + 1 = 242 + 1 = 243 \][/tex]
So, [tex]\( B = 243 \)[/tex].
For [tex]\( T_{20} \)[/tex]:
[tex]\[ T_{20} = 2 \cdot 20^2 + 1 = 2 \cdot 400 + 1 = 800 + 1 = 801 \][/tex]
So, [tex]\( C = 801 \)[/tex].
### Step 3: Determine the 17th Term
Now, let's find the 17th term [tex]\( T_{17} \)[/tex]:
[tex]\[ T_{17} = 2 \cdot 17^2 + 1 = 2 \cdot 289 + 1 = 578 + 1 = 579 \][/tex]
Thus, the 17th term is 579.
### Summary
- The value of [tex]\( d \)[/tex] is [tex]\( 2 \)[/tex].
- The values are [tex]\( A = 51 \)[/tex], [tex]\( B = 243 \)[/tex], and [tex]\( C = 801 \)[/tex].
- The 17th term is [tex]\( 579 \)[/tex].
