Solving Geometric Sequence Problems

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Solving Geometric Sequence Problems

For a geometric sequence, $T_n=ar^{n-1}$ . Ratios of separated terms eliminate $a$ : $T_q/T_p=r^{q-p}$ . Check whether negative ratios or even powers allow more than one algebraic candidate.

Worked reasoning

$T_5/T_2=r^{5-2}$ .
$162/6=27=r^3$ .
The stated positive ratio gives $r=3$ .

Key idea

$T_q/T_p=r^{q-p}$ connects separated terms in a geometric sequence.

A positive-ratio geometric sequence has $T_2=6$ and $T_5=162$ . Find $r$ .

Which statement best captures the central mathematical idea in Solving Geometric Sequence Problems?

$T_q/T_p=r^{q-p}$ connects separated terms in a geometric sequence.

Separated geometric terms differ by adding a fixed multiple of $r$ .

A memorised formula is valid even when its conditions are not met.

The diagram or notation replaces the need to justify each mathematical step.

When starting a problem about Solving Geometric Sequence Problems, which move is most reliable?

Divide two term equations to remove the common first-term factor.

Round every value before identifying the mathematical structure.

Choose the longest available formula and substitute immediately.

Assume the desired conclusion and work backwards without checking equivalence.

Which statement is a misconception that must be rejected when working with Solving Geometric Sequence Problems?

Separated geometric terms differ by adding a fixed multiple of $r$ .

$T_q/T_p=r^{q-p}$ connects separated terms in a geometric sequence.

Divide two term equations to remove the common first-term factor.

Substitute the chosen ratio back into both original term equations.

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