Find term n of a geometric sequence using the rule

A sequence starts at t₀ = 6, then gives t₁ = 12 and t₂ = 24. Predict t₃ if the same multiplier continues

The initial value is t₀ = 12. Move the index and the ratio. Compare a term with the next term, including ratios below 1 and exactly 1.

With R held fixed at 1.5, what stays constant as n changes?

From repeated multiplication to a power

A recurrence tells you how to get the next term. A direct rule gives any term from the starting value and the number of multiplications

1. 1Check consecutive ratios For nonzero terms, divide each term by the previous one. A geometric sequence has a constant ratio \(\displaystyle 12/6=24/12=2\)
2. 2State the starting index In this lesson the initial value is t₀. Term tₙ is reached after n multiplications \(\displaystyle t_0=6,\quad t_n=6R^n\)
3. 3Power only the ratio The initial value is used once. The ratio is applied repeatedly \(\displaystyle t_3=6\times2^3=48\)
4. 4Adjust for a first term labelled t₁ From term 1 to term n there are n−1 multiplications. Neither indexing convention is universal \(\displaystyle t_n=t_1R^{n-1}\)
5. 5Use exact supplied values Keep fractions and full calculator precision through the calculation. Round only when asked

Sequence

\(\displaystyle \begin{gathered}t_0=8\\ t_1=12\\ t_2=18\end{gathered}\)

Find the common ratio

Sequence

\(\displaystyle \begin{gathered}t_0=3\\ t_{n+1}=2t_n\end{gathered}\)

How many multiplications take the sequence from t₀ to t₅?

Sequence

\(\displaystyle \begin{gathered}t_0=3\\ t_{n+1}=2t_n\end{gathered}\)

Find t₅ using that rule

Sequence

\(\displaystyle \begin{gathered}t_0=160\\ t_{n+1}=\tfrac12t_n\end{gathered}\)

Now use a decaying sequence. Find t₄

A new sequence is labelled from t₁ = 3, with each term twice the previous term. Which rule gives t₅?

Sequence

\(\displaystyle \begin{gathered}t_3=54\\ R=3\end{gathered}\)

Recover the initial value from a later term and the ratio

Which positive sequence is geometric?

Sequence

\(\displaystyle \begin{gathered}t_0=80\\ R=1-0.25\end{gathered}\)

A model starts at 80 and loses 25% each period. Find its value after 3 periods

For a positive starting value and a ratio between zero and one, what does the mathematical model do as the index increases?

A population model doubles forever. Why should that long-term prediction be questioned?

Sequence

\(\displaystyle \begin{gathered}a_1=7\\ R=3\end{gathered}\)

Transfer: the first term is labelled a₁ = 7 and each next term is three times the previous one. Find a₄

Written rule check: signs

Use paper for this ungraded production task. Write both a complete recurrence and an explicit rule for each sequence before continuing to the worked self-check. Automatic practice in this topic still uses positive initial values and ratios

1. 1A: a negative initial value The initial index is 0, the starting value is −6 and each next term is twice the preceding term. Write both rules and check index 2
2. 2B: a negative ratio The initial index is 0, the starting value is 18 and each next term is negative one half times the preceding term. Write both rules and check index 3

Self-check: negative values and ratios

Compare your own equations with these rules. This comparison is not automatically graded. A correct later value alone does not replace writing both requested rules

1. 1A: use the negative start once At index 2 the value is −24. The ratio remains positive \(\displaystyle \begin{gathered}u_0=-6,\quad u_{n+1}=2u_n\\u_n=-6\times2^n\end{gathered}\)
2. 2B: power the entire negative ratio At index 3 the value is negative nine quarters. The minus sign belongs inside the powered base; it is not a fixed minus in front of every term \(\displaystyle \begin{gathered}u_0=18,\quad u_{n+1}=-\frac12u_n\\u_n=18\left(-\frac12\right)^n\\u_3=18\left(-\frac12\right)^3=18\left(-\frac18\right)=-\frac94\end{gathered}\)

Written rule check: zero cases

Use paper again. Write the complete recurrence and a rule valid at every non-negative integer index. Keep the initial value separate when necessary, then continue to compare

1. 1C: zero multiplier The value at index 0 is 7 and every next term is obtained by multiplying the previous term by 0. Your explicit description must also work at index 0 without using 0 to the power 0
2. 2D: zero initial value The value at index 0 is 0 and the specified multiplier is −2. Write both rules. Then decide whether the list 0,0,0 alone would identify that ratio uniquely

Self-check: keep zero boundaries explicit

These are ungraded written checks, not automatic proficiency evidence. Revise any equation that fails at index 0. Do not infer a unique multiplier by dividing zero by zero

1. 1C: separate index 0 from later indices The recurrence starts at 7; every later term is zero. A piecewise explicit rule avoids the undefined zero-to-the-power-zero issue \(\displaystyle \begin{gathered}u_0=7,\quad u_{n+1}=0\\u_n=\begin{cases}7&n=0\\0&n\ge1\end{cases}\end{gathered}\)
2. 2D: zero remains zero The specified multiplier −2 gives this recurrence, but the all-zero list also fits any other fixed multiplier. Its ratio cannot be recovered uniquely from the terms \(\displaystyle \begin{gathered}u_0=0,\quad u_{n+1}=-2u_n\\u_n=0\end{gathered}\)