To solve for the common ratio $r$ and the first term $a$ of the geometric sequence, we will use the general formula for the $n$-th term of a geometric sequence:

$T_n = a r^{n-1}.$

Given:

1. $T_{12} = a r^{11} = 1,417,176$,
2. $T_{17} = a r^{16} = -344,373,768$.

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Step 1: Form the ratio of $T_{17}$ to $T_{12}$

Divide the second equation by the first:

$\frac{T_{17}}{T_{12}} = \frac{a r^{16}}{a r^{11}} = r^{16-11}.$

Simplify:

$\frac{-344,373,768}{1,417,176} = r^5.$

Calculate $r^5$:

$r^5 = \frac{-344,373,768}{1,417,176} = -243.$

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Step 2: Solve for $r$

Take the fifth root of both sides:

$r = \sqrt[5]{-243} = -3.$

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Step 3: Solve for $a$

Substitute $r = -3$ into the first equation $T_{12} = a r^{11}$:

$1,417,176 = a (-3)^{11}.$

Evaluate $(-3)^{11}$:

$(-3)^{11} = -177,147.$

Thus:

$1,417,176 = a (-177,147).$

Solve for $a$:

$a = \frac{1,417,176}{-177,147} = -8.$

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Final Results:

• The first term: $a = -8$,
• The common ratio: $r = -3$.

Verification:

1. $T_{12} = -8(-3)^{11} = -8(-177,147) = 1,417,176$ ✅,
2. $T_{17} = -8(-3)^{16} = -8(43,046,721) = -344,373,768$ ✅.

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Related Questions:

1. What is the formula for the sum of the first $n$ terms of a geometric sequence?
2. How do you determine if a sequence is geometric?
3. Can a geometric sequence have a ratio $r = 1$? What would that mean?
4. What happens to a geometric sequence as $n \to \infty$ if $|r| < 1$?
5. How would you derive the $n$-th term formula from first principles?

Tip: In a geometric sequence, if $r < 0$, terms will alternate between positive and negative.