To find the greatest possible value for [tex]\( t_5 \)[/tex] in a geometric series where the first term [tex]\( t_1 = 12 \)[/tex] and the sum of the first three terms [tex]\( S_3 = 372 \)[/tex], follow these steps:
1. Express the Sum of the First Three Terms:
The sum of the first three terms of a geometric series is given by:
[tex]\[
S_3 = t_1 \left( \frac{1 - r^3}{1 - r} \right)
\][/tex]
Given [tex]\( S_3 = 372 \)[/tex] and [tex]\( t_1 = 12 \)[/tex], substitute these values into the equation:
[tex]\[
372 = 12 \left( \frac{1 - r^3}{1 - r} \right)
\][/tex]
2. Simplify the Equation:
Divide both sides of the equation by 12:
[tex]\[
31 = \frac{1 - r^3}{1 - r}
\][/tex]
Notice that [tex]\( \frac{1 - r^3}{1 - r} \)[/tex] is a well-known formula for the sum of a geometric series up to [tex]\( r^2 \)[/tex], which simplifies to:
[tex]\[
1 + r + r^2
\][/tex]
Therefore, we have:
[tex]\[
31 = 1 + r + r^2
\][/tex]
3. Form a Quadratic Equation:
Rearrange the equation into a standard quadratic form:
[tex]\[
r^2 + r - 30 = 0
\][/tex]
4. Solve the Quadratic Equation:
Use the quadratic formula:
[tex]\[
r = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\][/tex]
Here, [tex]\( a = 1 \)[/tex], [tex]\( b = 1 \)[/tex], and [tex]\( c = -30 \)[/tex]. Substitute these values into the formula:
[tex]\[
r = \frac{-1 \pm \sqrt{1^2 - 4 \cdot 1 \cdot (-30)}}{2 \cdot 1}
\][/tex]
[tex]\[
r = \frac{-1 \pm \sqrt{1 + 120}}{2}
\][/tex]
[tex]\[
r = \frac{-1 \pm \sqrt{121}}{2}
\][/tex]
[tex]\[
r = \frac{-1 \pm 11}{2}
\][/tex]
Thus, we get two roots:
[tex]\[
r_1 = \frac{-1 + 11}{2} = 5
\][/tex]
[tex]\[
r_2 = \frac{-1 - 11}{2} = -6
\][/tex]
5. Calculate the Fifth Term [tex]\( t_5 \)[/tex]:
The nth term of a geometric series is given by:
[tex]\[
t_n = t_1 \cdot r^{n-1}
\][/tex]
For [tex]\( t_5 \)[/tex]:
[tex]\[
t_5 = t_1 \cdot r^4
\][/tex]
Substitute [tex]\( t_1 = 12 \)[/tex] and the two values of [tex]\( r \)[/tex]:
- For [tex]\( r_1 = 5 \)[/tex]:
[tex]\[
t_5 = 12 \cdot 5^4 = 12 \cdot 625 = 7500
\][/tex]
- For [tex]\( r_2 = -6 \)[/tex]:
[tex]\[
t_5 = 12 \cdot (-6)^4 = 12 \cdot 1296 = 15552
\][/tex]
6. Identify the Greatest Possible Value:
Compare the two values of [tex]\( t_5 \)[/tex]:
[tex]\[
\text{Greatest possible value of } t_5 = \max(7500, 15552) = 15552
\][/tex]
Therefore, the greatest possible value for [tex]\( t_5 \)[/tex] is [tex]\( 15552 \)[/tex].
