Sure, let's go through this step-by-step.
### 3.1.1 Determine [tex]\( T_{22} \)[/tex]
Given the arithmetic sequence: 7, 10, 13, ...
In an arithmetic sequence, the general formula for the [tex]\( n \)[/tex]-th term [tex]\( T_n \)[/tex] is:
[tex]\[ T_n = a + (n - 1)d \][/tex]
where:
- [tex]\( a \)[/tex] is the first term,
- [tex]\( d \)[/tex] is the common difference,
- [tex]\( n \)[/tex] is the term number.
Here, we have:
- [tex]\( a = 7 \)[/tex]
- [tex]\( d = 3 \)[/tex]
- [tex]\( n = 22 \)[/tex]
Substituting these values into the formula:
[tex]\[ T_{22} = 7 + (22 - 1) \cdot 3 \][/tex]
[tex]\[ T_{22} = 7 + 21 \cdot 3 \][/tex]
[tex]\[ T_{22} = 7 + 63 \][/tex]
[tex]\[ T_{22} = 70 \][/tex]
So, [tex]\( T_{22} = 70 \)[/tex].
### 3.1.2 Determine the value of [tex]\( n \)[/tex] if [tex]\( S_n = 847 \)[/tex]
The sum of the first [tex]\( n \)[/tex] terms [tex]\( S_n \)[/tex] in an arithmetic sequence is given by:
[tex]\[ S_n = \frac{n}{2} \left( 2a + (n - 1)d \right) \][/tex]
We are given [tex]\( S_n = 847 \)[/tex] and need to find [tex]\( n \)[/tex]. Given the values:
- [tex]\( a = 7 \)[/tex]
- [tex]\( d = 3 \)[/tex]
- [tex]\( S_n = 847 \)[/tex]
Substitute these values into the sum formula:
[tex]\[ 847 = \frac{n}{2} \left( 2 \cdot 7 + (n - 1) \cdot 3 \right) \][/tex]
[tex]\[ 847 = \frac{n}{2} \left( 14 + 3n - 3 \right) \][/tex]
[tex]\[ 847 = \frac{n}{2} \left( 3n + 11 \right) \][/tex]
[tex]\[ 847 = \frac{3n^2 + 11n}{2} \][/tex]
Multiply both sides by 2 to clear the fraction:
[tex]\[ 1694 = 3n^2 + 11n \][/tex]
Rearrange the equation to standard quadratic form:
[tex]\[ 3n^2 + 11n - 1694 = 0 \][/tex]
We solve this quadratic equation using the quadratic formula [tex]\( n = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = 3 \)[/tex], [tex]\( b = 11 \)[/tex], and [tex]\( c = -1694 \)[/tex].
Calculating the discriminant:
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = 11^2 - 4 \cdot 3 \cdot (-1694) \][/tex]
[tex]\[ \Delta = 121 + 20328 \][/tex]
[tex]\[ \Delta = 20449 \][/tex]
Since the discriminant is positive, the quadratic equation has two real solutions:
[tex]\[ n = \frac{-11 \pm \sqrt{20449}}{6} \][/tex]
Calculating the square root and the solutions:
[tex]\[ \sqrt{20449} = 143 \][/tex]
[tex]\[ n = \frac{-11 + 143}{6} \][/tex]
[tex]\[ n = \frac{132}{6} \][/tex]
[tex]\[ n = 22 \][/tex]
[tex]\[ n = \frac{-11 - 143}{6} \][/tex]
[tex]\[ n = \frac{-154}{6} \][/tex]
[tex]\[ n = -25.6667 \][/tex] (which is not a valid solution since [tex]\( n \)[/tex] must be a positive integer)
Therefore, the valid value of [tex]\( n \)[/tex] is:
[tex]\[ n = 22 \][/tex]
So, the value of [tex]\( n \)[/tex] if [tex]\( S_n = 847 \)[/tex] is [tex]\( n = 22 \)[/tex].
